{"id":1741,"date":"2023-10-12T00:32:04","date_gmt":"2023-10-12T00:32:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-exponents-and-scientific-notation\/"},"modified":"2023-10-12T00:32:04","modified_gmt":"2023-10-12T00:32:04","slug":"introduction-exponents-and-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-exponents-and-scientific-notation\/","title":{"raw":"Exponents and Scientific Notation","rendered":"Exponents and Scientific Notation"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li class=\"li2\"><span class=\"s1\">Use the rules of exponents to simplify exponential expressions.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use scientific notation.<\/span><\/li>\n<\/ul>\n<\/div>\nMathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.\n\nUsing a calculator, we enter [latex]2,048\\times 1,536\\times 48\\times 24\\times 3,600[\/latex] and press ENTER. The calculator displays 1.304596316E13. What does this mean? The \"E13\" portion of the result represents the exponent 13 of ten, so there are a maximum of approximately [latex]1.3\\times {10}^{13}[\/latex] bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.\n<h2>Rules for Exponents<\/h2>\nConsider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, <em>x<\/em>, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.\n<div style=\"text-align: center\">[latex]\\begin{align}x^{3}\\cdot x^{4}&amp;=\\stackrel{\\text{3 factors }}{(x\\cdot x\\cdot x)} \\stackrel{\\text{ 4 factors}}{(x\\cdot x\\cdot x\\cdot x)} \\\\ &amp; =\\stackrel{7 \\text{ factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ &amp; =x^{7}\\end{align}[\/latex]<\/div>\nThe result is that [latex]{x}^{3}\\cdot {x}^{4}={x}^{3+4}={x}^{7}[\/latex].\n\nNotice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the <em>product rule of exponents.<\/em>\n<div style=\"text-align: center\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\nNow consider an example with real numbers.\n<div style=\"text-align: center\">[latex]{2}^{3}\\cdot {2}^{4}={2}^{3+4}={2}^{7}[\/latex]<\/div>\nWe can always check that this is true by simplifying each exponential expression. We find that [latex]{2}^{3}[\/latex] is 8, [latex]{2}^{4}[\/latex] is 16, and [latex]{2}^{7}[\/latex] is 128. The product [latex]8\\cdot 16[\/latex] equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.\n<div class=\"textbox\">\n<h3>A General Note: The Product Rule of Exponents<\/h3>\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], the product rule of exponents states that\n<div style=\"text-align: center\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\n \t<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"878162\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"878162\"]\nUse the product rule to simplify each expression.\n<ol>\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\n \t<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\nAt first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.\n<div style=\"text-align: center\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\nNotice we get the same result by adding the three exponents in one step.\n<div style=\"text-align: center\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]{k}^{6}\\cdot {k}^{9}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{4}\\cdot \\left(\\dfrac{2}{y}\\right)[\/latex]<\/li>\n \t<li>[latex]{t}^{3}\\cdot {t}^{6}\\cdot {t}^{5}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"562258\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"562258\"]\n<ol>\n \t<li>[latex]{k}^{15}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]{t}^{14}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h3>Using the Quotient Rule of Exponents<\/h3>\nThe <em>quotient rule of exponents<\/em> allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as [latex]\\dfrac{{y}^{m}}{{y}^{n}}[\/latex], where [latex]m&gt;n[\/latex]. Consider the example [latex]\\dfrac{{y}^{9}}{{y}^{5}}[\/latex]. Perform the division by canceling common factors.\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{y^{9}}{y^{5}} &amp;=\\frac{y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y}{y\\cdot y\\cdot y\\cdot y\\cdot y} \\\\[1mm] &amp;=\\frac{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot y\\cdot y\\cdot y\\cdot y}{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}} \\\\[1mm] &amp; =\\frac{y\\cdot y\\cdot y\\cdot y}{1} \\\\[1mm] &amp; =y^{4}\\\\ \\text{ }\\end{align}[\/latex]<\/div>\nNotice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.\n<div style=\"text-align: center\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\nIn other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.\n<div style=\"text-align: center\">[latex]\\dfrac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[\/latex]<\/div>\nFor the time being, we must be aware of the condition [latex]m&gt;n[\/latex]. Otherwise, the difference [latex]m-n[\/latex] could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m&gt;n[\/latex], the quotient rule of exponents states that\n<div style=\"text-align: center\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"717838\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"717838\"]\n\nUse the quotient rule to simplify each expression.\n<ol>\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]\\dfrac{{s}^{75}}{{s}^{68}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(-3\\right)}^{6}}{-3}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(e{f}^{2}\\right)}^{5}}{{\\left(e{f}^{2}\\right)}^{3}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"677916\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"677916\"]\n<ol>\n \t<li>[latex]{s}^{7}[\/latex]<\/li>\n \t<li>[latex]{\\left(-3\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]{\\left(e{f}^{2}\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Using the Power Rule of Exponents<\/h2>\nSuppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the <em>power rule of exponents<\/em>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({x}^{2}\\right)}^{3}&amp; = \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}} \\\\ &amp; = \\stackrel{{3\\text{ factors}}}{\\overbrace{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ &amp; = x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ &amp; = {x}^{6} \\end{align}[\/latex]<\/div>\nThe exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.\n<div style=\"text-align: center\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\nBe careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center\" colspan=\"5\">Product Rule<\/th>\n<th style=\"text-align: center\" colspan=\"6\">Power Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>&nbsp;[latex]5^{3+4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{3\\cdot4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{5+2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>&nbsp;[latex]x^{5\\cdot2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{10}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10} [\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7+1-} [\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(\\left(3a\\right)^{7}\\right)^{10} [\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7\\cdot10} [\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: The Power Rule of Exponents<\/h3>\nFor any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that\n<div style=\"text-align: center\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power Rule<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\n \t<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"992335\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"992335\"]\n\nUse the power rule to simplify each expression.\n<ol>\n \t<li>[latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\n \t<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\n \t<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each of the following products with a single base. Do not simplify further.\n<ol>\n \t<li>[latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\n \t<li>[latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"875151\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"875151\"]\n<ol>\n \t<li>[latex]{\\left(3y\\right)}^{24}[\/latex]<\/li>\n \t<li>[latex]{t}^{35}[\/latex]<\/li>\n \t<li>[latex]{\\left(-g\\right)}^{16}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&amp;theme=oea&amp;iframe_resize_id=mom80\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93399&amp;theme=oea&amp;iframe_resize_id=mom90\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&amp;theme=oea&amp;iframe_resize_id=mom100\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\nThe following video gives more examples of using the power rule to simplify expressions with exponents.\n\nhttps:\/\/youtu.be\/VjcKU5rA7F8\n<h2>Zero and Negative Exponents<\/h2>\nReturn to the quotient rule. We made the condition that [latex]m&gt;n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the <em>zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.\n<p style=\"text-align: center\">[latex]\\dfrac{t^{8}}{t^{8}}=\\dfrac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\nIf we were to simplify the original expression using the quotient rule, we would have\n<div style=\"text-align: center\">[latex]\\dfrac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\nIf we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.\n<div style=\"text-align: center\">[latex]{a}^{0}=1[\/latex]<\/div>\nThe sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.\n<div class=\"textbox\">\n<h3>A General Note: The Zero Exponent Rule of Exponents<\/h3>\nFor any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that\n<div style=\"text-align: center\">[latex]{a}^{0}=1[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Zero Exponent Rule<\/h3>\nSimplify each expression using the zero exponent rule of exponents.\n<ol>\n \t<li>[latex]\\dfrac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"913171\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"913171\"]\nUse the zero exponent and other rules to simplify each expression.\n<ol>\n \t<li>[latex]\\begin{align}\\frac{c^{3}}{c^{3}} &amp; =c^{3-3} \\\\ &amp; =c^{0} \\\\ &amp; =1\\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} \\frac{-3{x}^{5}}{{x}^{5}}&amp; = -3\\cdot \\frac{{x}^{5}}{{x}^{5}} \\\\ &amp; = -3\\cdot {x}^{5 - 5} \\\\ &amp; = -3\\cdot {x}^{0} \\\\ &amp; = -3\\cdot 1 \\\\ &amp; = -3 \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}&amp; = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}} &amp;&amp; \\text{Use the product rule in the denominator}. \\\\ &amp; = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}} &amp;&amp; \\text{Simplify}. \\\\ &amp; = {\\left({j}^{2}k\\right)}^{4 - 4} &amp;&amp; \\text{Use the quotient rule}. \\\\ &amp; = {\\left({j}^{2}k\\right)}^{0} &amp;&amp; \\text{Simplify}. \\\\ &amp; = 1 \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}&amp; = 5{\\left(r{s}^{2}\\right)}^{2 - 2} &amp;&amp; \\text{Use the quotient rule}. \\\\ &amp; = 5{\\left(r{s}^{2}\\right)}^{0} &amp;&amp; \\text{Simplify}. \\\\ &amp; = 5\\cdot 1 &amp;&amp; \\text{Use the zero exponent rule}. \\\\ &amp; = 5 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify each expression using the zero exponent rule of exponents.\n<ol>\n \t<li>[latex]\\dfrac{{t}^{7}}{{t}^{7}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\cdot {t}^{5}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"703483\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"703483\"]\n<ol>\n \t<li>[latex]1[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n \t<li>[latex]1[\/latex]<\/li>\n \t<li>[latex]1[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=44120&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7833&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\nIn this video we show more examples of how to simplify expressions with zero exponents.\n\nhttps:\/\/youtu.be\/rpoUg32utlc\n<h2>Using the Negative Rule of Exponents<\/h2>\nAnother useful result occurs if we relax the condition that [latex]m&gt;n[\/latex] in the quotient rule even further. For example, can we simplify [latex]\\dfrac{{h}^{3}}{{h}^{5}}[\/latex]? When [latex]m&lt;n[\/latex]\u2014that is, where the difference [latex]m-n[\/latex] is negative\u2014we can use the <em>negative rule of exponents<\/em> to simplify the expression to its reciprocal.\n\nDivide one exponential expression by another with a larger exponent. Use our example, [latex]\\dfrac{{h}^{3}}{{h}^{5}}[\/latex].\n<div style=\"text-align: center\">[latex]\\begin{align} \\frac{{h}^{3}}{{h}^{5}}&amp; = \\frac{h\\cdot h\\cdot h}{h\\cdot h\\cdot h\\cdot h\\cdot h} \\\\ &amp; = \\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot h\\cdot h} \\\\ &amp; = \\frac{1}{h\\cdot h} \\\\ &amp; = \\frac{1}{{h}^{2}} \\end{align}[\/latex]<\/div>\nIf we were to simplify the original expression using the quotient rule, we would have\n<div style=\"text-align: center\">[latex]\\begin{align} \\frac{{h}^{3}}{{h}^{5}}&amp; = {h}^{3 - 5} \\\\ &amp; = {h}^{-2} \\end{align}[\/latex]<\/div>\nPutting the answers together, we have [latex]{h}^{-2}=\\dfrac{1}{{h}^{2}}[\/latex]. This is true for any nonzero real number, or any variable representing a nonzero real number.\n\nA factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar\u2014from numerator to denominator or vice versa.\n<div style=\"text-align: center\">[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/div>\nWe have shown that the exponential expression [latex]{a}^{n}[\/latex] is defined when [latex]n[\/latex] is a natural number, 0, or the negative of a natural number. That means that [latex]{a}^{n}[\/latex] is defined for any integer [latex]n[\/latex]. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer [latex]n[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: The Negative Rule of Exponents<\/h3>\nFor any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that\n<div style=\"text-align: center\">[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Negative Exponent Rule<\/h3>\nWrite each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.\n<ol>\n \t<li>[latex]\\dfrac{{\\theta }^{3}}{{\\theta }^{10}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{z}^{2}\\cdot z}{{z}^{4}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(-5{t}^{3}\\right)}^{4}}{{\\left(-5{t}^{3}\\right)}^{8}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"746940\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"746940\"]\n<ol>\n \t<li>[latex]\\dfrac{{\\theta }^{3}}{{\\theta }^{10}}={\\theta }^{3 - 10}={\\theta }^{-7}=\\dfrac{1}{{\\theta }^{7}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{z}^{2}\\cdot z}{{z}^{4}}=\\dfrac{{z}^{2+1}}{{z}^{4}}=\\dfrac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\\dfrac{1}{z}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(-5{t}^{3}\\right)}^{4}}{{\\left(-5{t}^{3}\\right)}^{8}}={\\left(-5{t}^{3}\\right)}^{4 - 8}={\\left(-5{t}^{3}\\right)}^{-4}=\\dfrac{1}{{\\left(-5{t}^{3}\\right)}^{4}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.\n<ol>\n \t<li>[latex]\\frac{{\\left(-3t\\right)}^{2}}{{\\left(-3t\\right)}^{8}}[\/latex]<\/li>\n \t<li>[latex]\\frac{{f}^{47}}{{f}^{49}\\cdot f}[\/latex]<\/li>\n \t<li>[latex]\\frac{2{k}^{4}}{5{k}^{7}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"85205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"85205\"]\n<ol>\n \t<li>[latex]\\frac{1}{{\\left(-3t\\right)}^{6}}[\/latex]<\/li>\n \t<li>[latex]\\frac{1}{{f}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\frac{2}{5{k}^{3}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=51959&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109762&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109765&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\nIn the following video we show more examples of how to find the power of a quotient.\n\nhttps:\/\/youtu.be\/BoBe31pRxFM\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product and Quotient Rules<\/h3>\nWrite each of the following products with a single base. Do not simplify further. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{b}^{2}\\cdot {b}^{-8}[\/latex]<\/li>\n \t<li>[latex]{\\left(-x\\right)}^{5}\\cdot {\\left(-x\\right)}^{-5}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{-7z}{{\\left(-7z\\right)}^{5}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"803639\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"803639\"]\n<ol>\n \t<li>[latex]{b}^{2}\\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\\frac{1}{{b}^{6}}[\/latex]<\/li>\n \t<li>[latex]{\\left(-x\\right)}^{5}\\cdot {\\left(-x\\right)}^{-5}={\\left(-x\\right)}^{5 - 5}={\\left(-x\\right)}^{0}=1[\/latex]<\/li>\n \t<li>[latex]\\dfrac{-7z}{{\\left(-7z\\right)}^{5}}=\\dfrac{{\\left(-7z\\right)}^{1}}{{\\left(-7z\\right)}^{5}}={\\left(-7z\\right)}^{1 - 5}={\\left(-7z\\right)}^{-4}=\\dfrac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each of the following products with a single base. Do not simplify further. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{t}^{-11}\\cdot {t}^{6}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{25}^{12}}{{25}^{13}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"592096\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"592096\"]\n<ol>\n \t<li>[latex]{t}^{-5}=\\dfrac{1}{{t}^{5}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{25}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93393&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h2>Finding the Power of a Product<\/h2>\nTo simplify the power of a product of two exponential expressions, we can use the <em>power of a product rule of exponents,<\/em> which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider [latex]{\\left(pq\\right)}^{3}[\/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors.\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left(pq\\right)}^{3}&amp; = \\stackrel{3\\text{ factors}}{{\\left(pq\\right)\\cdot \\left(pq\\right)\\cdot \\left(pq\\right)}} \\\\ &amp; = p\\cdot q\\cdot p\\cdot q\\cdot p\\cdot q \\\\ &amp; = \\stackrel{3\\text{ factors}}{{p\\cdot p\\cdot p}}\\cdot \\stackrel{3\\text{ factors}}{{q\\cdot q\\cdot q}} \\\\ &amp; = {p}^{3}\\cdot {q}^{3}\\hfill \\end{align}[\/latex]<\/div>\nIn other words, [latex]{\\left(pq\\right)}^{3}={p}^{3}\\cdot {q}^{3}[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: The Power of a Product Rule of Exponents<\/h3>\nFor any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a product rule of exponents states that\n<div style=\"text-align: center\">[latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power of a Product Rule<\/h3>\nSimplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{\\left(a{b}^{2}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left(2t\\right)}^{15}[\/latex]<\/li>\n \t<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\n \t<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"189543\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"189543\"]\nUse the product and quotient rules and the new definitions to simplify each expression.\n<ol>\n \t<li>[latex]{\\left(a{b}^{2}\\right)}^{3}={\\left(a\\right)}^{3}\\cdot {\\left({b}^{2}\\right)}^{3}={a}^{1\\cdot 3}\\cdot {b}^{2\\cdot 3}={a}^{3}{b}^{6}[\/latex]<\/li>\n \t<li>[latex]2{t}^{15}={\\left(2\\right)}^{15}\\cdot {\\left(t\\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[\/latex]<\/li>\n \t<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}={\\left(-2\\right)}^{3}\\cdot {\\left({w}^{3}\\right)}^{3}=-8\\cdot {w}^{3\\cdot 3}=-8{w}^{9}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{\\left(-7z\\right)}^{4}}=\\dfrac{1}{{\\left(-7\\right)}^{4}\\cdot {\\left(z\\right)}^{4}}=\\dfrac{1}{2,401{z}^{4}}[\/latex]<\/li>\n \t<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}={\\left({e}^{-2}\\right)}^{7}\\cdot {\\left({f}^{2}\\right)}^{7}={e}^{-2\\cdot 7}\\cdot {f}^{2\\cdot 7}={e}^{-14}{f}^{14}=\\dfrac{{f}^{14}}{{e}^{14}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{\\left({g}^{2}{h}^{3}\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]{\\left(5t\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left(-3{y}^{5}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{\\left({a}^{6}{b}^{7}\\right)}^{3}}[\/latex]<\/li>\n \t<li>[latex]{\\left({r}^{3}{s}^{-2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"540817\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"540817\"]\n<ol>\n \t<li>[latex]{g}^{10}{h}^{15}[\/latex]<\/li>\n \t<li>[latex]125{t}^{3}[\/latex]<\/li>\n \t<li>[latex]-27{y}^{15}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{a}^{18}{b}^{21}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{r}^{12}}{{s}^{8}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14047&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14058&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14059&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\nIn the following video we show more examples of how to find hte power of a product.\n\nhttps:\/\/youtu.be\/p-2UkpJQWpo\n<h2>Finding the Power of a Quotient<\/h2>\nTo simplify the power of a quotient of two expressions, we can use the <em>power of a quotient rule,<\/em> which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let\u2019s look at the following example.\n<div style=\"text-align: center\">[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}=\\dfrac{{f}^{14}}{{e}^{14}}[\/latex]<\/div>\nLet\u2019s rewrite the original problem differently and look at the result.\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({e}^{-2}{f}^{2}\\right)}^{7}&amp; = {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7} \\\\ &amp; = \\frac{{f}^{14}}{{e}^{14}} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\nIt appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({e}^{-2}{f}^{2}\\right)}^{7}&amp; = {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7} \\\\ &amp; = \\frac{{\\left({f}^{2}\\right)}^{7}}{{\\left({e}^{2}\\right)}^{7}} \\\\ &amp; = \\frac{{f}^{2\\cdot 7}}{{e}^{2\\cdot 7}} \\\\ &amp; = \\frac{{f}^{14}}{{e}^{14}} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: The Power of a Quotient Rule of Exponents<\/h3>\nFor any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a quotient rule of exponents states that\n<div style=\"text-align: center\">[latex]\\large{\\left(\\frac{a}{b}\\right)}^{n}=\\frac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power of a Quotient Rule<\/h3>\nSimplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{\\left(\\dfrac{4}{{z}^{11}}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{p}{{q}^{3}}\\right)}^{6}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{-1}{{t}^{2}}\\right)}^{27}[\/latex]<\/li>\n \t<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}[\/latex]<\/li>\n \t<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"835767\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"835767\"]\n<ol>\n \t<li>[latex]{\\left(\\dfrac{4}{{z}^{11}}\\right)}^{3}=\\dfrac{{\\left(4\\right)}^{3}}{{\\left({z}^{11}\\right)}^{3}}=\\dfrac{64}{{z}^{11\\cdot 3}}=\\dfrac{64}{{z}^{33}}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{p}{{q}^{3}}\\right)}^{6}=\\dfrac{{\\left(p\\right)}^{6}}{{\\left({q}^{3}\\right)}^{6}}=\\dfrac{{p}^{1\\cdot 6}}{{q}^{3\\cdot 6}}=\\dfrac{{p}^{6}}{{q}^{18}}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{-1}{{t}^{2}}\\right)}^{27}=\\dfrac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\\dfrac{-1}{{t}^{2\\cdot 27}}=\\dfrac{-1}{{t}^{54}}=-\\dfrac{1}{{t}^{54}}[\/latex]<\/li>\n \t<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}={\\left(\\dfrac{{j}^{3}}{{k}^{2}}\\right)}^{4}=\\dfrac{{\\left({j}^{3}\\right)}^{4}}{{\\left({k}^{2}\\right)}^{4}}=\\dfrac{{j}^{3\\cdot 4}}{{k}^{2\\cdot 4}}=\\dfrac{{j}^{12}}{{k}^{8}}[\/latex]<\/li>\n \t<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}={\\left(\\dfrac{1}{{m}^{2}{n}^{2}}\\right)}^{3}=\\dfrac{{\\left(1\\right)}^{3}}{{\\left({m}^{2}{n}^{2}\\right)}^{3}}=\\dfrac{1}{{\\left({m}^{2}\\right)}^{3}{\\left({n}^{2}\\right)}^{3}}=\\dfrac{1}{{m}^{2\\cdot 3}\\cdot {n}^{2\\cdot 3}}=\\dfrac{1}{{m}^{6}{n}^{6}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.\n<ol>\n \t<li>[latex]{\\left(\\dfrac{{b}^{5}}{c}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{5}{{u}^{8}}\\right)}^{4}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{-1}{{w}^{3}}\\right)}^{35}[\/latex]<\/li>\n \t<li>[latex]{\\left({p}^{-4}{q}^{3}\\right)}^{8}[\/latex]<\/li>\n \t<li>[latex]{\\left({c}^{-5}{d}^{-3}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"815731\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"815731\"]\n<ol>\n \t<li>[latex]\\dfrac{{b}^{15}}{{c}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{625}{{u}^{32}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{-1}{{w}^{105}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{q}^{24}}{{p}^{32}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{c}^{20}{d}^{12}}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14046&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14051&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=43231&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Simplifying Exponential Expressions<\/h2>\nRecall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Exponential Expressions<\/h3>\nSimplify each expression and write the answer with positive exponents only.\n<ol>\n \t<li>[latex]{\\left(6{m}^{2}{n}^{-1}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]{17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}[\/latex]<\/li>\n \t<li>[latex]\\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)[\/latex]<\/li>\n \t<li>[latex]{\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"9384\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"9384\"]\n<ol>\n \t<li>[latex]\\begin{align} {\\left(6{m}^{2}{n}^{-1}\\right)}^{3}&amp; = {\\left(6\\right)}^{3}{\\left({m}^{2}\\right)}^{3}{\\left({n}^{-1}\\right)}^{3}&amp;&amp; \\text{The power of a product rule} \\\\ &amp; = {6}^{3}{m}^{2\\cdot 3}{n}^{-1\\cdot 3}&amp;&amp; \\text{The power rule}\\hfill \\\\ &amp; = 216{m}^{6}{n}^{-3}&amp;&amp; \\text{Simplify}. \\\\ &amp; = \\frac{216{m}^{6}}{{n}^{3}}&amp;&amp; \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} {17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}&amp; =&amp; {17}^{5 - 4-3}&amp;&amp; \\text{The product rule}\\hfill \\\\ &amp; = {17}^{-2}&amp;&amp; \\text{Simplify}. \\\\ &amp; = \\frac{1}{{17}^{2}}\\text{ or }\\frac{1}{289}&amp;&amp; \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} {\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}&amp; = \\frac{{\\left({u}^{-1}v\\right)}^{2}}{{\\left({v}^{-1}\\right)}^{2}}&amp;&amp; \\text{The power of a quotient rule} \\\\ &amp; = \\frac{{u}^{-2}{v}^{2}}{{v}^{-2}}&amp;&amp; \\text{The power of a product rule} \\\\ &amp; = {u}^{-2}{v}^{2-\\left(-2\\right)}&amp;&amp; \\text{The quotient rule} \\\\ &amp; = {u}^{-2}{v}^{4}&amp;&amp; \\text{Simplify}. \\\\ &amp; = \\frac{{v}^{4}}{{u}^{2}}&amp;&amp; \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} \\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)&amp; =&amp; -2\\cdot 5\\cdot {a}^{3}\\cdot {a}^{-2}\\cdot {b}^{-1}\\cdot {b}^{2}&amp;&amp; \\text{Commutative and associative laws of multiplication} \\\\ &amp; = -10\\cdot {a}^{3 - 2}\\cdot {b}^{-1+2}&amp;&amp; \\text{The product rule} \\\\ &amp; = -10ab&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} {\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}&amp; = {\\left({x}^{2}\\sqrt{2}\\right)}^{4 - 4} &amp;&amp; \\text{The product rule} \\\\ &amp; = {\\left({x}^{2}\\sqrt{2}\\right)}^{0}&amp;&amp; \\text{Simplify}. \\\\ &amp; = 1&amp;&amp; \\text{The zero exponent rule} \\end{align}[\/latex]<\/li>\n \t<li>[latex]\\begin{align} \\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}&amp; = \\frac{{\\left(3\\right)}^{5}\\cdot {\\left({w}^{2}\\right)}^{5}}{{\\left(6\\right)}^{2}\\cdot {\\left({w}^{-2}\\right)}^{2}}&amp;&amp; \\text{The power of a product rule} \\\\ &amp; = \\frac{{3}^{5}{w}^{2\\cdot 5}}{{6}^{2}{w}^{-2\\cdot 2}}&amp;&amp; \\text{The power rule} \\\\ &amp; = \\frac{243{w}^{10}}{36{w}^{-4}} &amp;&amp; \\text{Simplify}. \\\\ &amp; = \\frac{27{w}^{10-\\left(-4\\right)}}{4}&amp;&amp; \\text{The quotient rule and reduce fraction} \\\\ &amp; = \\frac{27{w}^{14}}{4}&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify each expression and write the answer with positive exponents only.\n<ol>\n \t<li>[latex]{\\left(2u{v}^{-2}\\right)}^{-3}[\/latex]<\/li>\n \t<li>[latex]{x}^{8}\\cdot {x}^{-12}\\cdot x[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{{e}^{2}{f}^{-3}}{{f}^{-1}}\\right)}^{2}[\/latex]<\/li>\n \t<li>[latex]\\left(9{r}^{-5}{s}^{3}\\right)\\left(3{r}^{6}{s}^{-4}\\right)[\/latex]<\/li>\n \t<li>[latex]{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{-3}{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{3}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{\\left(2{h}^{2}k\\right)}^{4}}{{\\left(7{h}^{-1}{k}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"31244\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"31244\"]\n<ol>\n \t<li>[latex]\\dfrac{{v}^{6}}{8{u}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{1}{{x}^{3}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{{e}^{4}}{{f}^{4}}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{27r}{s}[\/latex]<\/li>\n \t<li>[latex]1[\/latex]<\/li>\n \t<li>[latex]\\dfrac{16{h}^{10}}{49}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14056&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14057&amp;theme=oea&amp;iframe_resize_id=mom80\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14060&amp;theme=oea&amp;iframe_resize_id=mom90\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=43896&amp;theme=oea&amp;iframe_resize_id=mom95\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\nIn the following video we show more examples of how to find the power of a quotient.\nhttps:\/\/youtu.be\/BoBe31pRxFM\n<h2>Scientific Notation<\/h2>\nRecall at the beginning of the section that we found the number [latex]1.3\\times {10}^{13}[\/latex] when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?\n\nA shorthand method of writing very small and very large numbers is called <strong>scientific notation<\/strong>, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places <em>n<\/em> that you moved the decimal point. Multiply the decimal number by 10 raised to a power of <em>n<\/em>. If you moved the decimal left as in a very large number, [latex]n[\/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[\/latex] is negative.\n\nFor example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225857\/CNX_CAT_Figure_01_02_001.jpg\" alt=\"The number 2,780,418 is written with an arrow extending to another number: 2.780418. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 6 places left.\">\n\nWe obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.\n<div style=\"text-align: center\">[latex]2.780418\\times {10}^{6}[\/latex]<\/div>\nWorking with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225859\/CNX_CAT_Figure_01_02_002.jpg\" alt=\"The number 0.00000000000047 is written with an arrow extending to another number: 00000000000004.7. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 13 places right.\">\n\nBe careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.\n<div style=\"text-align: center\">[latex]4.7\\times {10}^{-13}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Scientific Notation<\/h3>\nA number is written in <strong>scientific notation<\/strong> if it is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|&lt;10[\/latex] and [latex]n[\/latex] is an integer.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Converting Standard Notation to Scientific Notation<\/h3>\nWrite each number in scientific notation.\n<ol>\n \t<li>Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m<\/li>\n \t<li>Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m<\/li>\n \t<li>Number of stars in Andromeda Galaxy: 1,000,000,000,000<\/li>\n \t<li>Diameter of electron: 0.00000000000094 m<\/li>\n \t<li>Probability of being struck by lightning in any single year: 0.00000143<\/li>\n<\/ol>\n[reveal-answer q=\"955934\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"955934\"]\n<p style=\"text-align: left\">1.\n[latex]\\begin{align}&amp;\\underset{\\leftarrow 22\\text{ places}}{{24,000,000,000,000,000,000,000\\text{ m}}} \\\\ &amp;2.4\\times {10}^{22}\\text{ m} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">2.\n[latex]\\begin{align}&amp;\\underset{\\leftarrow 21\\text{ places}}{{1,300,000,000,000,000,000,000\\text{ m}}} \\\\ &amp;1.3\\times {10}^{21}\\text{ m} \\\\ &amp;\\text{ } \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">3.\n[latex]\\begin{align}&amp;\\underset{\\leftarrow 12\\text{ places}}{{1,000,000,000,000}} \\\\ &amp;1\\times {10}^{12} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">4.\n[latex]\\begin{align}&amp;\\underset{\\rightarrow 6\\text{ places}}{{0.00000000000094\\text{ m}}} \\\\ &amp;9.4\\times {10}^{-13}\\text{ m} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">5.\n[latex]\\begin{align}\\underset{\\to 6\\text{ places}}{{0.00000143}} \\\\ 1.43\\times {10}^{-6} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nObserve that, if the given number is greater than 1, as in examples a\u2013c, the exponent of 10 is positive; and if the number is less than 1, as in examples d\u2013e, the exponent is negative.\n\n[\/hidden-answer]\n\n<\/div>\n<div><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite each number in scientific notation.\n<ol>\n \t<li>U.S. national debt per taxpayer (April 2014): $152,000<\/li>\n \t<li>World population (April 2014): 7,158,000,000<\/li>\n \t<li>World gross national income (April 2014): $85,500,000,000,000<\/li>\n \t<li>Time for light to travel 1 m: 0.00000000334 s<\/li>\n \t<li>Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715<\/li>\n<\/ol>\n[reveal-answer q=\"745560\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"745560\"]\n<ol>\n \t<li>[latex]$1.52\\times {10}^{5}[\/latex]<\/li>\n \t<li>[latex]7.158\\times {10}^{9}[\/latex]<\/li>\n \t<li>[latex]$8.55\\times {10}^{13}[\/latex]<\/li>\n \t<li>[latex]3.34\\times {10}^{-9}[\/latex]<\/li>\n \t<li>[latex]7.15\\times {10}^{-8}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2466&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2836&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h2>Converting from Scientific to Standard Notation<\/h2>\nTo convert a number in <strong>scientific notation<\/strong> to standard notation, simply reverse the process. Move the decimal [latex]n[\/latex] places to the right if [latex]n[\/latex] is positive or [latex]n[\/latex] places to the left if [latex]n[\/latex] is negative and add zeros as needed. Remember, if [latex]n[\/latex] is positive, the value of the number is greater than 1, and if [latex]n[\/latex] is negative, the value of the number is less than one.\n<div class=\"textbox exercises\">\n<h3>Example: Converting Scientific Notation to Standard Notation<\/h3>\nConvert each number in scientific notation to standard notation.\n<ol>\n \t<li>[latex]3.547\\times {10}^{14}[\/latex]<\/li>\n \t<li>[latex]-2\\times {10}^{6}[\/latex]<\/li>\n \t<li>[latex]7.91\\times {10}^{-7}[\/latex]<\/li>\n \t<li>[latex]-8.05\\times {10}^{-12}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"545940\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"545940\"]\n\n1.\n[latex]\\begin{align}&amp;3.547\\times {10}^{14} \\\\ &amp;\\underset{\\to 14\\text{ places}}{{3.54700000000000}} \\\\ &amp;354,700,000,000,000 \\\\ \\text{ }\\end{align}[\/latex]\n\n2.\n[latex]\\begin{align}&amp;-2\\times {10}^{6} \\\\ &amp;\\underset{\\to 6\\text{ places}}{{-2.000000}} \\\\ &amp;-2,000,000 \\\\ \\text{ }\\end{align}[\/latex]\n\n3.\n[latex]\\begin{align}&amp;7.91\\times {10}^{-7} \\\\ &amp;\\underset{\\to 7\\text{ places}}{{0000007.91}} \\\\ &amp;0.000000791 \\\\ \\text{ }\\end{align}[\/latex]\n\n4.\n[latex]\\begin{align}&amp;-8.05\\times {10}^{-12} \\\\ &amp;\\underset{\\to 12\\text{ places}}{{-000000000008.05}} \\\\ &amp;-0.00000000000805 \\\\ \\text{ }\\end{align}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nConvert each number in scientific notation to standard notation.\n<ol>\n \t<li>[latex]7.03\\times {10}^{5}[\/latex]<\/li>\n \t<li>[latex]-8.16\\times {10}^{11}[\/latex]<\/li>\n \t<li>[latex]-3.9\\times {10}^{-13}[\/latex]<\/li>\n \t<li>[latex]8\\times {10}^{-6}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"655272\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"655272\"]\n<ol>\n \t<li>[latex]703,000[\/latex]<\/li>\n \t<li>[latex]-816,000,000,000[\/latex]<\/li>\n \t<li>[latex]-0.00000000000039[\/latex]<\/li>\n \t<li>[latex]0.000008[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2839&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2841&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2874&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h2>Using Scientific Notation in Applications<\/h2>\nScientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and 1 L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops. Therefore, there are approximately [latex]3\\cdot \\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!\n\nWhen performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product [latex]\\left(7\\times {10}^{4}\\right)\\cdot \\left(5\\times {10}^{6}\\right)=35\\times {10}^{10}[\/latex]. The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as [latex]3.5\\times 10[\/latex]. That adds a ten to the exponent of the answer.\n<div style=\"text-align: center\">[latex]\\left(35\\right)\\times {10}^{10}=\\left(3.5\\times 10\\right)\\times {10}^{10}=3.5\\times \\left(10\\times {10}^{10}\\right)=3.5\\times {10}^{11}[\/latex]<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Scientific Notation<\/h3>\nPerform the operations and write the answer in scientific notation.\n<ol>\n \t<li>[latex]\\left(8.14\\times {10}^{-7}\\right)\\left(6.5\\times {10}^{10}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"761197\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"761197\"]\n\n1.\n[latex]\\begin{align}\\left(8.14 \\times 10^{-7}\\right)\\left(6.5 \\times 10^{10}\\right) &amp; =\\left(8.14 \\times 6.5\\right)\\left(10^{-7} \\times 10^{10}\\right) &amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; =\\left(52.91\\right)\\left(10^{3}\\right) &amp;&amp; \\text{Product rule of exponents} \\\\ &amp; =5.291 \\times 10^{4} &amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]\n\n2.\n[latex]\\begin{align} \\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)&amp; = \\left(\\frac{4}{-1.52}\\right)\\left(\\frac{{10}^{5}}{{10}^{9}}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(-2.63\\right)\\left({10}^{-4}\\right)&amp;&amp; \\text{Quotient rule of exponents} \\\\ &amp; = -2.63\\times {10}^{-4}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]\n\n3.\n[latex]\\begin{align} \\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)&amp; = \\left(2.7\\times 6.04\\right)\\left({10}^{5}\\times {10}^{13}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(16.308\\right)\\left({10}^{18}\\right)&amp;&amp; \\text{Product rule of exponents} \\\\ &amp; = 1.6308\\times {10}^{19}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]\n\n4.\n[latex]\\begin{align} \\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)&amp; = \\left(\\frac{1.2}{9.6}\\right)\\left(\\frac{{10}^{8}}{{10}^{5}}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(0.125\\right)\\left({10}^{3}\\right)&amp;&amp; \\text{Quotient rule of exponents} \\\\ &amp; = 1.25\\times {10}^{2}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]\n\n5.\n[latex]\\begin{align} \\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)&amp; = \\left[3.33\\times \\left(-1.05\\right)\\times 5.62\\right]\\left({10}^{4}\\times {10}^{7}\\times {10}^{5}\\right) \\\\ &amp; \\approx \\left(-19.65\\right)\\left({10}^{16}\\right) \\\\ &amp; = -1.965\\times {10}^{17} \\end{align}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nWatch the following video to see more examples of writing numbers in scientific notation.\nhttps:\/\/youtu.be\/fsNu3AdIgdk\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nPerform the operations and write the answer in scientific notation.\n<ol>\n \t<li>[latex]\\left(-7.5\\times {10}^{8}\\right)\\left(1.13\\times {10}^{-2}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(1.24\\times {10}^{11}\\right)\\div \\left(1.55\\times {10}^{18}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(3.72\\times {10}^{9}\\right)\\left(8\\times {10}^{3}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(9.933\\times {10}^{23}\\right)\\div \\left(-2.31\\times {10}^{17}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(-6.04\\times {10}^{9}\\right)\\left(7.3\\times {10}^{2}\\right)\\left(-2.81\\times {10}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"578130\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"578130\"]\n<ol>\n \t<li>[latex]-8.475\\times {10}^{6}[\/latex]<\/li>\n \t<li>[latex]8\\times {10}^{-8}[\/latex]<\/li>\n \t<li>[latex]2.976\\times {10}^{13}[\/latex]<\/li>\n \t<li>[latex]-4.3\\times {10}^{6}[\/latex]<\/li>\n \t<li>[latex]\\approx 1.24\\times {10}^{15}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2824&amp;theme=oea&amp;iframe_resize_id=mom3 0\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2828&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2830&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Scientific Notation to Solve Problems<\/h3>\nIn April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.\n\n[reveal-answer q=\"431376\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"431376\"]\nThe population was [latex]308,000,000=3.08\\times {10}^{8}[\/latex].\n\nThe national debt was [latex]\\$ 17,547,000,000,000 \\approx \\$1.75 \\times 10^{13}[\/latex].\n\nTo find the amount of debt per citizen, divide the national debt by the number of citizens.\n<div style=\"text-align: center\">[latex]\\begin{align} \\left(1.75\\times {10}^{13}\\right)\\div \\left(3.08\\times {10}^{8}\\right)&amp; = \\left(\\frac{1.75}{3.08}\\right)\\cdot \\left(\\frac{{10}^{13}}{{10}^{8}}\\right) \\\\ &amp; \\approx 0.57\\times {10}^{5}\\hfill \\\\ &amp; = 5.7\\times {10}^{4} \\end{align}[\/latex]<\/div>\nThe debt per citizen at the time was about [latex]\\$5.7\\times {10}^{4}[\/latex], or $57,000.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAn average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.\n\n[reveal-answer q=\"985422\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"985422\"]\n\nNumber of cells: [latex]3\\times {10}^{13}[\/latex]; length of a cell: [latex]8\\times {10}^{-6}[\/latex] m; total length: [latex]2.4\\times {10}^{8}[\/latex] m or [latex]240,000,000[\/latex] m.\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3295&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=101856&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"650\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=102452&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"550\"><\/iframe>\n\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong>Rules of Exponents<\/strong>\nFor nonzero real numbers [latex]a[\/latex] and [latex]b[\/latex] and integers [latex]m[\/latex] and [latex]n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Product rule<\/strong><\/td>\n<td>[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Quotient rule<\/strong><\/td>\n<td>[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power rule<\/strong><\/td>\n<td>[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Zero exponent rule<\/strong><\/td>\n<td>[latex]{a}^{0}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Negative rule<\/strong><\/td>\n<td>[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power of a product rule<\/strong><\/td>\n<td>[latex]{\\left(a\\cdot b\\right)}^{n}={a}^{n}\\cdot {b}^{n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power of a quotient rule<\/strong><\/td>\n<td>[latex]{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Products of exponential expressions with the same base can be simplified by adding exponents.<\/li>\n \t<li>Quotients of exponential expressions with the same base can be simplified by subtracting exponents.<\/li>\n \t<li>Powers of exponential expressions with the same base can be simplified by multiplying exponents.<\/li>\n \t<li>An expression with exponent zero is defined as 1.<\/li>\n \t<li>An expression with a negative exponent is defined as a reciprocal.<\/li>\n \t<li>The power of a product of factors is the same as the product of the powers of the same factors.<\/li>\n \t<li>The power of a quotient of factors is the same as the quotient of the powers of the same factors.<\/li>\n \t<li>The rules for exponential expressions can be combined to simplify more complicated expressions.<\/li>\n \t<li>Scientific notation uses powers of 10 to simplify very large or very small numbers.<\/li>\n \t<li>Scientific notation may be used to simplify calculations with very large or very small numbers.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>scientific notation&nbsp;<\/strong>a shorthand notation for writing very large or very small numbers in the form [latex]a\\times {10}^{n}[\/latex] where [latex]1\\le |a|&lt;10[\/latex] and [latex]n[\/latex] is an integer\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Use the rules of exponents to simplify exponential expressions.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use scientific notation.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.<\/p>\n<p>Using a calculator, we enter [latex]2,048\\times 1,536\\times 48\\times 24\\times 3,600[\/latex] and press ENTER. The calculator displays 1.304596316E13. What does this mean? The &#8220;E13&#8221; portion of the result represents the exponent 13 of ten, so there are a maximum of approximately [latex]1.3\\times {10}^{13}[\/latex] bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.<\/p>\n<h2>Rules for Exponents<\/h2>\n<p>Consider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, <em>x<\/em>, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}x^{3}\\cdot x^{4}&=\\stackrel{\\text{3 factors }}{(x\\cdot x\\cdot x)} \\stackrel{\\text{ 4 factors}}{(x\\cdot x\\cdot x\\cdot x)} \\\\ & =\\stackrel{7 \\text{ factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ & =x^{7}\\end{align}[\/latex]<\/div>\n<p>The result is that [latex]{x}^{3}\\cdot {x}^{4}={x}^{3+4}={x}^{7}[\/latex].<\/p>\n<p>Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the <em>product rule of exponents.<\/em><\/p>\n<div style=\"text-align: center\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<p>Now consider an example with real numbers.<\/p>\n<div style=\"text-align: center\">[latex]{2}^{3}\\cdot {2}^{4}={2}^{3+4}={2}^{7}[\/latex]<\/div>\n<p>We can always check that this is true by simplifying each exponential expression. We find that [latex]{2}^{3}[\/latex] is 8, [latex]{2}^{4}[\/latex] is 16, and [latex]{2}^{7}[\/latex] is 128. The product [latex]8\\cdot 16[\/latex] equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Product Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], the product rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\n<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q878162\">Show Solution<\/span><\/p>\n<div id=\"q878162\" class=\"hidden-answer\" style=\"display: none\">\nUse the product rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\n<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n<p>At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.<\/p>\n<div style=\"text-align: center\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\n<p>Notice we get the same result by adding the three exponents in one step.<\/p>\n<div style=\"text-align: center\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]{k}^{6}\\cdot {k}^{9}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{4}\\cdot \\left(\\dfrac{2}{y}\\right)[\/latex]<\/li>\n<li>[latex]{t}^{3}\\cdot {t}^{6}\\cdot {t}^{5}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q562258\">Show Solution<\/span><\/p>\n<div id=\"q562258\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{k}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{t}^{14}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Using the Quotient Rule of Exponents<\/h3>\n<p>The <em>quotient rule of exponents<\/em> allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as [latex]\\dfrac{{y}^{m}}{{y}^{n}}[\/latex], where [latex]m>n[\/latex]. Consider the example [latex]\\dfrac{{y}^{9}}{{y}^{5}}[\/latex]. Perform the division by canceling common factors.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{y^{9}}{y^{5}} &=\\frac{y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y}{y\\cdot y\\cdot y\\cdot y\\cdot y} \\\\[1mm] &=\\frac{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot y\\cdot y\\cdot y\\cdot y}{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}} \\\\[1mm] & =\\frac{y\\cdot y\\cdot y\\cdot y}{1} \\\\[1mm] & =y^{4}\\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.<\/p>\n<div style=\"text-align: center\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<p>In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.<\/p>\n<div style=\"text-align: center\">[latex]\\dfrac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[\/latex]<\/div>\n<p>For the time being, we must be aware of the condition [latex]m>n[\/latex]. Otherwise, the difference [latex]m-n[\/latex] could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m>n[\/latex], the quotient rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717838\">Show Solution<\/span><\/p>\n<div id=\"q717838\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the quotient rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]\\dfrac{{s}^{75}}{{s}^{68}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(-3\\right)}^{6}}{-3}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(e{f}^{2}\\right)}^{5}}{{\\left(e{f}^{2}\\right)}^{3}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q677916\">Show Solution<\/span><\/p>\n<div id=\"q677916\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{s}^{7}[\/latex]<\/li>\n<li>[latex]{\\left(-3\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(e{f}^{2}\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Power Rule of Exponents<\/h2>\n<p>Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the <em>power rule of exponents<\/em>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({x}^{2}\\right)}^{3}& = \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}} \\\\ & = \\stackrel{{3\\text{ factors}}}{\\overbrace{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ & = x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ & = {x}^{6} \\end{align}[\/latex]<\/div>\n<p>The exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.<\/p>\n<div style=\"text-align: center\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<p>Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center\" colspan=\"5\">Product Rule<\/th>\n<th style=\"text-align: center\" colspan=\"6\">Power Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>&nbsp;[latex]5^{3+4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{3\\cdot4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{5+2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>&nbsp;[latex]x^{5\\cdot2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{10}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7+1-}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(\\left(3a\\right)^{7}\\right)^{10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7\\cdot10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: The Power Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992335\">Show Solution<\/span><\/p>\n<div id=\"q992335\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the power rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875151\">Show Solution<\/span><\/p>\n<div id=\"q875151\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{\\left(3y\\right)}^{24}[\/latex]<\/li>\n<li>[latex]{t}^{35}[\/latex]<\/li>\n<li>[latex]{\\left(-g\\right)}^{16}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&amp;theme=oea&amp;iframe_resize_id=mom80\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93399&amp;theme=oea&amp;iframe_resize_id=mom90\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&amp;theme=oea&amp;iframe_resize_id=mom100\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>The following video gives more examples of using the power rule to simplify expressions with exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions Using the Power Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VjcKU5rA7F8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Zero and Negative Exponents<\/h2>\n<p>Return to the quotient rule. We made the condition that [latex]m>n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the <em>zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{t^{8}}{t^{8}}=\\dfrac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\n<p>If we were to simplify the original expression using the quotient rule, we would have<\/p>\n<div style=\"text-align: center\">[latex]\\dfrac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\n<p>If we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.<\/p>\n<div style=\"text-align: center\">[latex]{a}^{0}=1[\/latex]<\/div>\n<p>The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Zero Exponent Rule of Exponents<\/h3>\n<p>For any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]{a}^{0}=1[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Zero Exponent Rule<\/h3>\n<p>Simplify each expression using the zero exponent rule of exponents.<\/p>\n<ol>\n<li>[latex]\\dfrac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q913171\">Show Solution<\/span><\/p>\n<div id=\"q913171\" class=\"hidden-answer\" style=\"display: none\">\nUse the zero exponent and other rules to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\begin{align}\\frac{c^{3}}{c^{3}} & =c^{3-3} \\\\ & =c^{0} \\\\ & =1\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{-3{x}^{5}}{{x}^{5}}& = -3\\cdot \\frac{{x}^{5}}{{x}^{5}} \\\\ & = -3\\cdot {x}^{5 - 5} \\\\ & = -3\\cdot {x}^{0} \\\\ & = -3\\cdot 1 \\\\ & = -3 \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}& = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}} && \\text{Use the product rule in the denominator}. \\\\ & = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}} && \\text{Simplify}. \\\\ & = {\\left({j}^{2}k\\right)}^{4 - 4} && \\text{Use the quotient rule}. \\\\ & = {\\left({j}^{2}k\\right)}^{0} && \\text{Simplify}. \\\\ & = 1 \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}& = 5{\\left(r{s}^{2}\\right)}^{2 - 2} && \\text{Use the quotient rule}. \\\\ & = 5{\\left(r{s}^{2}\\right)}^{0} && \\text{Simplify}. \\\\ & = 5\\cdot 1 && \\text{Use the zero exponent rule}. \\\\ & = 5 && \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each expression using the zero exponent rule of exponents.<\/p>\n<ol>\n<li>[latex]\\dfrac{{t}^{7}}{{t}^{7}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\cdot {t}^{5}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703483\">Show Solution<\/span><\/p>\n<div id=\"q703483\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=44120&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7833&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In this video we show more examples of how to simplify expressions with zero exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Expressions Using the Quotient and Zero Exponent Rules\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rpoUg32utlc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Negative Rule of Exponents<\/h2>\n<p>Another useful result occurs if we relax the condition that [latex]m>n[\/latex] in the quotient rule even further. For example, can we simplify [latex]\\dfrac{{h}^{3}}{{h}^{5}}[\/latex]? When [latex]m<n[\/latex]\u2014that is, where the difference [latex]m-n[\/latex] is negative\u2014we can use the <em>negative rule of exponents<\/em> to simplify the expression to its reciprocal.<\/p>\n<p>Divide one exponential expression by another with a larger exponent. Use our example, [latex]\\dfrac{{h}^{3}}{{h}^{5}}[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} \\frac{{h}^{3}}{{h}^{5}}& = \\frac{h\\cdot h\\cdot h}{h\\cdot h\\cdot h\\cdot h\\cdot h} \\\\ & = \\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot h\\cdot h} \\\\ & = \\frac{1}{h\\cdot h} \\\\ & = \\frac{1}{{h}^{2}} \\end{align}[\/latex]<\/div>\n<p>If we were to simplify the original expression using the quotient rule, we would have<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} \\frac{{h}^{3}}{{h}^{5}}& = {h}^{3 - 5} \\\\ & = {h}^{-2} \\end{align}[\/latex]<\/div>\n<p>Putting the answers together, we have [latex]{h}^{-2}=\\dfrac{1}{{h}^{2}}[\/latex]. This is true for any nonzero real number, or any variable representing a nonzero real number.<\/p>\n<p>A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar\u2014from numerator to denominator or vice versa.<\/p>\n<div style=\"text-align: center\">[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/div>\n<p>We have shown that the exponential expression [latex]{a}^{n}[\/latex] is defined when [latex]n[\/latex] is a natural number, 0, or the negative of a natural number. That means that [latex]{a}^{n}[\/latex] is defined for any integer [latex]n[\/latex]. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer [latex]n[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Negative Rule of Exponents<\/h3>\n<p>For any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Negative Exponent Rule<\/h3>\n<p>Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\theta }^{3}}{{\\theta }^{10}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{z}^{2}\\cdot z}{{z}^{4}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(-5{t}^{3}\\right)}^{4}}{{\\left(-5{t}^{3}\\right)}^{8}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q746940\">Show Solution<\/span><\/p>\n<div id=\"q746940\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{{\\theta }^{3}}{{\\theta }^{10}}={\\theta }^{3 - 10}={\\theta }^{-7}=\\dfrac{1}{{\\theta }^{7}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{z}^{2}\\cdot z}{{z}^{4}}=\\dfrac{{z}^{2+1}}{{z}^{4}}=\\dfrac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\\dfrac{1}{z}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(-5{t}^{3}\\right)}^{4}}{{\\left(-5{t}^{3}\\right)}^{8}}={\\left(-5{t}^{3}\\right)}^{4 - 8}={\\left(-5{t}^{3}\\right)}^{-4}=\\dfrac{1}{{\\left(-5{t}^{3}\\right)}^{4}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]\\frac{{\\left(-3t\\right)}^{2}}{{\\left(-3t\\right)}^{8}}[\/latex]<\/li>\n<li>[latex]\\frac{{f}^{47}}{{f}^{49}\\cdot f}[\/latex]<\/li>\n<li>[latex]\\frac{2{k}^{4}}{5{k}^{7}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q85205\">Show Solution<\/span><\/p>\n<div id=\"q85205\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\frac{1}{{\\left(-3t\\right)}^{6}}[\/latex]<\/li>\n<li>[latex]\\frac{1}{{f}^{3}}[\/latex]<\/li>\n<li>[latex]\\frac{2}{5{k}^{3}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=51959&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109762&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109765&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to find the power of a quotient.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Expressions Using Exponent Rules (Power of a Quotient)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BoBe31pRxFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product and Quotient Rules<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{b}^{2}\\cdot {b}^{-8}[\/latex]<\/li>\n<li>[latex]{\\left(-x\\right)}^{5}\\cdot {\\left(-x\\right)}^{-5}[\/latex]<\/li>\n<li>[latex]\\dfrac{-7z}{{\\left(-7z\\right)}^{5}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803639\">Show Solution<\/span><\/p>\n<div id=\"q803639\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{b}^{2}\\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\\frac{1}{{b}^{6}}[\/latex]<\/li>\n<li>[latex]{\\left(-x\\right)}^{5}\\cdot {\\left(-x\\right)}^{-5}={\\left(-x\\right)}^{5 - 5}={\\left(-x\\right)}^{0}=1[\/latex]<\/li>\n<li>[latex]\\dfrac{-7z}{{\\left(-7z\\right)}^{5}}=\\dfrac{{\\left(-7z\\right)}^{1}}{{\\left(-7z\\right)}^{5}}={\\left(-7z\\right)}^{1 - 5}={\\left(-7z\\right)}^{-4}=\\dfrac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{t}^{-11}\\cdot {t}^{6}[\/latex]<\/li>\n<li>[latex]\\dfrac{{25}^{12}}{{25}^{13}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q592096\">Show Solution<\/span><\/p>\n<div id=\"q592096\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{t}^{-5}=\\dfrac{1}{{t}^{5}}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{25}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93393&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Power of a Product<\/h2>\n<p>To simplify the power of a product of two exponential expressions, we can use the <em>power of a product rule of exponents,<\/em> which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider [latex]{\\left(pq\\right)}^{3}[\/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left(pq\\right)}^{3}& = \\stackrel{3\\text{ factors}}{{\\left(pq\\right)\\cdot \\left(pq\\right)\\cdot \\left(pq\\right)}} \\\\ & = p\\cdot q\\cdot p\\cdot q\\cdot p\\cdot q \\\\ & = \\stackrel{3\\text{ factors}}{{p\\cdot p\\cdot p}}\\cdot \\stackrel{3\\text{ factors}}{{q\\cdot q\\cdot q}} \\\\ & = {p}^{3}\\cdot {q}^{3}\\hfill \\end{align}[\/latex]<\/div>\n<p>In other words, [latex]{\\left(pq\\right)}^{3}={p}^{3}\\cdot {q}^{3}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Power of a Product Rule of Exponents<\/h3>\n<p>For any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a product rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power of a Product Rule<\/h3>\n<p>Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left(a{b}^{2}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(2t\\right)}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\n<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q189543\">Show Solution<\/span><\/p>\n<div id=\"q189543\" class=\"hidden-answer\" style=\"display: none\">\nUse the product and quotient rules and the new definitions to simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left(a{b}^{2}\\right)}^{3}={\\left(a\\right)}^{3}\\cdot {\\left({b}^{2}\\right)}^{3}={a}^{1\\cdot 3}\\cdot {b}^{2\\cdot 3}={a}^{3}{b}^{6}[\/latex]<\/li>\n<li>[latex]2{t}^{15}={\\left(2\\right)}^{15}\\cdot {\\left(t\\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}={\\left(-2\\right)}^{3}\\cdot {\\left({w}^{3}\\right)}^{3}=-8\\cdot {w}^{3\\cdot 3}=-8{w}^{9}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{\\left(-7z\\right)}^{4}}=\\dfrac{1}{{\\left(-7\\right)}^{4}\\cdot {\\left(z\\right)}^{4}}=\\dfrac{1}{2,401{z}^{4}}[\/latex]<\/li>\n<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}={\\left({e}^{-2}\\right)}^{7}\\cdot {\\left({f}^{2}\\right)}^{7}={e}^{-2\\cdot 7}\\cdot {f}^{2\\cdot 7}={e}^{-14}{f}^{14}=\\dfrac{{f}^{14}}{{e}^{14}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left({g}^{2}{h}^{3}\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(5t\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(-3{y}^{5}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{\\left({a}^{6}{b}^{7}\\right)}^{3}}[\/latex]<\/li>\n<li>[latex]{\\left({r}^{3}{s}^{-2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q540817\">Show Solution<\/span><\/p>\n<div id=\"q540817\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{g}^{10}{h}^{15}[\/latex]<\/li>\n<li>[latex]125{t}^{3}[\/latex]<\/li>\n<li>[latex]-27{y}^{15}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{a}^{18}{b}^{21}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{r}^{12}}{{s}^{8}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14047&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14058&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14059&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to find hte power of a product.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Simplify Expressions Using Exponent Rules (Power of a Product)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/p-2UkpJQWpo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding the Power of a Quotient<\/h2>\n<p>To simplify the power of a quotient of two expressions, we can use the <em>power of a quotient rule,<\/em> which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let\u2019s look at the following example.<\/p>\n<div style=\"text-align: center\">[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}=\\dfrac{{f}^{14}}{{e}^{14}}[\/latex]<\/div>\n<p>Let\u2019s rewrite the original problem differently and look at the result.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({e}^{-2}{f}^{2}\\right)}^{7}& = {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7} \\\\ & = \\frac{{f}^{14}}{{e}^{14}} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<p>It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} {\\left({e}^{-2}{f}^{2}\\right)}^{7}& = {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7} \\\\ & = \\frac{{\\left({f}^{2}\\right)}^{7}}{{\\left({e}^{2}\\right)}^{7}} \\\\ & = \\frac{{f}^{2\\cdot 7}}{{e}^{2\\cdot 7}} \\\\ & = \\frac{{f}^{14}}{{e}^{14}} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: The Power of a Quotient Rule of Exponents<\/h3>\n<p>For any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a quotient rule of exponents states that<\/p>\n<div style=\"text-align: center\">[latex]\\large{\\left(\\frac{a}{b}\\right)}^{n}=\\frac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power of a Quotient Rule<\/h3>\n<p>Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left(\\dfrac{4}{{z}^{11}}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{p}{{q}^{3}}\\right)}^{6}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{-1}{{t}^{2}}\\right)}^{27}[\/latex]<\/li>\n<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}[\/latex]<\/li>\n<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q835767\">Show Solution<\/span><\/p>\n<div id=\"q835767\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{\\left(\\dfrac{4}{{z}^{11}}\\right)}^{3}=\\dfrac{{\\left(4\\right)}^{3}}{{\\left({z}^{11}\\right)}^{3}}=\\dfrac{64}{{z}^{11\\cdot 3}}=\\dfrac{64}{{z}^{33}}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{p}{{q}^{3}}\\right)}^{6}=\\dfrac{{\\left(p\\right)}^{6}}{{\\left({q}^{3}\\right)}^{6}}=\\dfrac{{p}^{1\\cdot 6}}{{q}^{3\\cdot 6}}=\\dfrac{{p}^{6}}{{q}^{18}}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{-1}{{t}^{2}}\\right)}^{27}=\\dfrac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\\dfrac{-1}{{t}^{2\\cdot 27}}=\\dfrac{-1}{{t}^{54}}=-\\dfrac{1}{{t}^{54}}[\/latex]<\/li>\n<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}={\\left(\\dfrac{{j}^{3}}{{k}^{2}}\\right)}^{4}=\\dfrac{{\\left({j}^{3}\\right)}^{4}}{{\\left({k}^{2}\\right)}^{4}}=\\dfrac{{j}^{3\\cdot 4}}{{k}^{2\\cdot 4}}=\\dfrac{{j}^{12}}{{k}^{8}}[\/latex]<\/li>\n<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}={\\left(\\dfrac{1}{{m}^{2}{n}^{2}}\\right)}^{3}=\\dfrac{{\\left(1\\right)}^{3}}{{\\left({m}^{2}{n}^{2}\\right)}^{3}}=\\dfrac{1}{{\\left({m}^{2}\\right)}^{3}{\\left({n}^{2}\\right)}^{3}}=\\dfrac{1}{{m}^{2\\cdot 3}\\cdot {n}^{2\\cdot 3}}=\\dfrac{1}{{m}^{6}{n}^{6}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left(\\dfrac{{b}^{5}}{c}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{5}{{u}^{8}}\\right)}^{4}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{-1}{{w}^{3}}\\right)}^{35}[\/latex]<\/li>\n<li>[latex]{\\left({p}^{-4}{q}^{3}\\right)}^{8}[\/latex]<\/li>\n<li>[latex]{\\left({c}^{-5}{d}^{-3}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815731\">Show Solution<\/span><\/p>\n<div id=\"q815731\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{{b}^{15}}{{c}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{625}{{u}^{32}}[\/latex]<\/li>\n<li>[latex]\\dfrac{-1}{{w}^{105}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{q}^{24}}{{p}^{32}}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{c}^{20}{d}^{12}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14046&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14051&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=43231&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Simplifying Exponential Expressions<\/h2>\n<p>Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Exponential Expressions<\/h3>\n<p>Simplify each expression and write the answer with positive exponents only.<\/p>\n<ol>\n<li>[latex]{\\left(6{m}^{2}{n}^{-1}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}[\/latex]<\/li>\n<li>[latex]\\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)[\/latex]<\/li>\n<li>[latex]{\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q9384\">Show Solution<\/span><\/p>\n<div id=\"q9384\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{align} {\\left(6{m}^{2}{n}^{-1}\\right)}^{3}& = {\\left(6\\right)}^{3}{\\left({m}^{2}\\right)}^{3}{\\left({n}^{-1}\\right)}^{3}&& \\text{The power of a product rule} \\\\ & = {6}^{3}{m}^{2\\cdot 3}{n}^{-1\\cdot 3}&& \\text{The power rule}\\hfill \\\\ & = 216{m}^{6}{n}^{-3}&& \\text{Simplify}. \\\\ & = \\frac{216{m}^{6}}{{n}^{3}}&& \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} {17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}& =& {17}^{5 - 4-3}&& \\text{The product rule}\\hfill \\\\ & = {17}^{-2}&& \\text{Simplify}. \\\\ & = \\frac{1}{{17}^{2}}\\text{ or }\\frac{1}{289}&& \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} {\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}& = \\frac{{\\left({u}^{-1}v\\right)}^{2}}{{\\left({v}^{-1}\\right)}^{2}}&& \\text{The power of a quotient rule} \\\\ & = \\frac{{u}^{-2}{v}^{2}}{{v}^{-2}}&& \\text{The power of a product rule} \\\\ & = {u}^{-2}{v}^{2-\\left(-2\\right)}&& \\text{The quotient rule} \\\\ & = {u}^{-2}{v}^{4}&& \\text{Simplify}. \\\\ & = \\frac{{v}^{4}}{{u}^{2}}&& \\text{The negative exponent rule} \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)& =& -2\\cdot 5\\cdot {a}^{3}\\cdot {a}^{-2}\\cdot {b}^{-1}\\cdot {b}^{2}&& \\text{Commutative and associative laws of multiplication} \\\\ & = -10\\cdot {a}^{3 - 2}\\cdot {b}^{-1+2}&& \\text{The product rule} \\\\ & = -10ab&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} {\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}& = {\\left({x}^{2}\\sqrt{2}\\right)}^{4 - 4} && \\text{The product rule} \\\\ & = {\\left({x}^{2}\\sqrt{2}\\right)}^{0}&& \\text{Simplify}. \\\\ & = 1&& \\text{The zero exponent rule} \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}& = \\frac{{\\left(3\\right)}^{5}\\cdot {\\left({w}^{2}\\right)}^{5}}{{\\left(6\\right)}^{2}\\cdot {\\left({w}^{-2}\\right)}^{2}}&& \\text{The power of a product rule} \\\\ & = \\frac{{3}^{5}{w}^{2\\cdot 5}}{{6}^{2}{w}^{-2\\cdot 2}}&& \\text{The power rule} \\\\ & = \\frac{243{w}^{10}}{36{w}^{-4}} && \\text{Simplify}. \\\\ & = \\frac{27{w}^{10-\\left(-4\\right)}}{4}&& \\text{The quotient rule and reduce fraction} \\\\ & = \\frac{27{w}^{14}}{4}&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each expression and write the answer with positive exponents only.<\/p>\n<ol>\n<li>[latex]{\\left(2u{v}^{-2}\\right)}^{-3}[\/latex]<\/li>\n<li>[latex]{x}^{8}\\cdot {x}^{-12}\\cdot x[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{{e}^{2}{f}^{-3}}{{f}^{-1}}\\right)}^{2}[\/latex]<\/li>\n<li>[latex]\\left(9{r}^{-5}{s}^{3}\\right)\\left(3{r}^{6}{s}^{-4}\\right)[\/latex]<\/li>\n<li>[latex]{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{-3}{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(2{h}^{2}k\\right)}^{4}}{{\\left(7{h}^{-1}{k}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q31244\">Show Solution<\/span><\/p>\n<div id=\"q31244\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{{v}^{6}}{8{u}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{{x}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{e}^{4}}{{f}^{4}}[\/latex]<\/li>\n<li>[latex]\\dfrac{27r}{s}[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]\\dfrac{16{h}^{10}}{49}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14056&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14057&amp;theme=oea&amp;iframe_resize_id=mom80\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14060&amp;theme=oea&amp;iframe_resize_id=mom90\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=43896&amp;theme=oea&amp;iframe_resize_id=mom95\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to find the power of a quotient.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-5\" title=\"Simplify Expressions Using Exponent Rules (Power of a Quotient)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BoBe31pRxFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Scientific Notation<\/h2>\n<p>Recall at the beginning of the section that we found the number [latex]1.3\\times {10}^{13}[\/latex] when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?<\/p>\n<p>A shorthand method of writing very small and very large numbers is called <strong>scientific notation<\/strong>, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places <em>n<\/em> that you moved the decimal point. Multiply the decimal number by 10 raised to a power of <em>n<\/em>. If you moved the decimal left as in a very large number, [latex]n[\/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[\/latex] is negative.<\/p>\n<p>For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225857\/CNX_CAT_Figure_01_02_001.jpg\" alt=\"The number 2,780,418 is written with an arrow extending to another number: 2.780418. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 6 places left.\" \/><\/p>\n<p>We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.<\/p>\n<div style=\"text-align: center\">[latex]2.780418\\times {10}^{6}[\/latex]<\/div>\n<p>Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225859\/CNX_CAT_Figure_01_02_002.jpg\" alt=\"The number 0.00000000000047 is written with an arrow extending to another number: 00000000000004.7. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 13 places right.\" \/><\/p>\n<p>Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.<\/p>\n<div style=\"text-align: center\">[latex]4.7\\times {10}^{-13}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Scientific Notation<\/h3>\n<p>A number is written in <strong>scientific notation<\/strong> if it is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|<10[\/latex] and [latex]n[\/latex] is an integer.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Converting Standard Notation to Scientific Notation<\/h3>\n<p>Write each number in scientific notation.<\/p>\n<ol>\n<li>Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m<\/li>\n<li>Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m<\/li>\n<li>Number of stars in Andromeda Galaxy: 1,000,000,000,000<\/li>\n<li>Diameter of electron: 0.00000000000094 m<\/li>\n<li>Probability of being struck by lightning in any single year: 0.00000143<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955934\">Show Solution<\/span><\/p>\n<div id=\"q955934\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left\">1.<br \/>\n[latex]\\begin{align}&\\underset{\\leftarrow 22\\text{ places}}{{24,000,000,000,000,000,000,000\\text{ m}}} \\\\ &2.4\\times {10}^{22}\\text{ m} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">2.<br \/>\n[latex]\\begin{align}&\\underset{\\leftarrow 21\\text{ places}}{{1,300,000,000,000,000,000,000\\text{ m}}} \\\\ &1.3\\times {10}^{21}\\text{ m} \\\\ &\\text{ } \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">3.<br \/>\n[latex]\\begin{align}&\\underset{\\leftarrow 12\\text{ places}}{{1,000,000,000,000}} \\\\ &1\\times {10}^{12} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">4.<br \/>\n[latex]\\begin{align}&\\underset{\\rightarrow 6\\text{ places}}{{0.00000000000094\\text{ m}}} \\\\ &9.4\\times {10}^{-13}\\text{ m} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p style=\"text-align: left\">5.<br \/>\n[latex]\\begin{align}\\underset{\\to 6\\text{ places}}{{0.00000143}} \\\\ 1.43\\times {10}^{-6} \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Observe that, if the given number is greater than 1, as in examples a\u2013c, the exponent of 10 is positive; and if the number is less than 1, as in examples d\u2013e, the exponent is negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each number in scientific notation.<\/p>\n<ol>\n<li>U.S. national debt per taxpayer (April 2014): $152,000<\/li>\n<li>World population (April 2014): 7,158,000,000<\/li>\n<li>World gross national income (April 2014): $85,500,000,000,000<\/li>\n<li>Time for light to travel 1 m: 0.00000000334 s<\/li>\n<li>Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q745560\">Show Solution<\/span><\/p>\n<div id=\"q745560\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]$1.52\\times {10}^{5}[\/latex]<\/li>\n<li>[latex]7.158\\times {10}^{9}[\/latex]<\/li>\n<li>[latex]$8.55\\times {10}^{13}[\/latex]<\/li>\n<li>[latex]3.34\\times {10}^{-9}[\/latex]<\/li>\n<li>[latex]7.15\\times {10}^{-8}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2466&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2836&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Converting from Scientific to Standard Notation<\/h2>\n<p>To convert a number in <strong>scientific notation<\/strong> to standard notation, simply reverse the process. Move the decimal [latex]n[\/latex] places to the right if [latex]n[\/latex] is positive or [latex]n[\/latex] places to the left if [latex]n[\/latex] is negative and add zeros as needed. Remember, if [latex]n[\/latex] is positive, the value of the number is greater than 1, and if [latex]n[\/latex] is negative, the value of the number is less than one.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Converting Scientific Notation to Standard Notation<\/h3>\n<p>Convert each number in scientific notation to standard notation.<\/p>\n<ol>\n<li>[latex]3.547\\times {10}^{14}[\/latex]<\/li>\n<li>[latex]-2\\times {10}^{6}[\/latex]<\/li>\n<li>[latex]7.91\\times {10}^{-7}[\/latex]<\/li>\n<li>[latex]-8.05\\times {10}^{-12}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q545940\">Show Solution<\/span><\/p>\n<div id=\"q545940\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<br \/>\n[latex]\\begin{align}&3.547\\times {10}^{14} \\\\ &\\underset{\\to 14\\text{ places}}{{3.54700000000000}} \\\\ &354,700,000,000,000 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align}&-2\\times {10}^{6} \\\\ &\\underset{\\to 6\\text{ places}}{{-2.000000}} \\\\ &-2,000,000 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>3.<br \/>\n[latex]\\begin{align}&7.91\\times {10}^{-7} \\\\ &\\underset{\\to 7\\text{ places}}{{0000007.91}} \\\\ &0.000000791 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>4.<br \/>\n[latex]\\begin{align}&-8.05\\times {10}^{-12} \\\\ &\\underset{\\to 12\\text{ places}}{{-000000000008.05}} \\\\ &-0.00000000000805 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert each number in scientific notation to standard notation.<\/p>\n<ol>\n<li>[latex]7.03\\times {10}^{5}[\/latex]<\/li>\n<li>[latex]-8.16\\times {10}^{11}[\/latex]<\/li>\n<li>[latex]-3.9\\times {10}^{-13}[\/latex]<\/li>\n<li>[latex]8\\times {10}^{-6}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q655272\">Show Solution<\/span><\/p>\n<div id=\"q655272\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]703,000[\/latex]<\/li>\n<li>[latex]-816,000,000,000[\/latex]<\/li>\n<li>[latex]-0.00000000000039[\/latex]<\/li>\n<li>[latex]0.000008[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2839&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2841&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2874&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Using Scientific Notation in Applications<\/h2>\n<p>Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and 1 L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops. Therefore, there are approximately [latex]3\\cdot \\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!<\/p>\n<p>When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product [latex]\\left(7\\times {10}^{4}\\right)\\cdot \\left(5\\times {10}^{6}\\right)=35\\times {10}^{10}[\/latex]. The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as [latex]3.5\\times 10[\/latex]. That adds a ten to the exponent of the answer.<\/p>\n<div style=\"text-align: center\">[latex]\\left(35\\right)\\times {10}^{10}=\\left(3.5\\times 10\\right)\\times {10}^{10}=3.5\\times \\left(10\\times {10}^{10}\\right)=3.5\\times {10}^{11}[\/latex]<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Scientific Notation<\/h3>\n<p>Perform the operations and write the answer in scientific notation.<\/p>\n<ol>\n<li>[latex]\\left(8.14\\times {10}^{-7}\\right)\\left(6.5\\times {10}^{10}\\right)[\/latex]<\/li>\n<li>[latex]\\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)[\/latex]<\/li>\n<li>[latex]\\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)[\/latex]<\/li>\n<li>[latex]\\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)[\/latex]<\/li>\n<li>[latex]\\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q761197\">Show Solution<\/span><\/p>\n<div id=\"q761197\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<br \/>\n[latex]\\begin{align}\\left(8.14 \\times 10^{-7}\\right)\\left(6.5 \\times 10^{10}\\right) & =\\left(8.14 \\times 6.5\\right)\\left(10^{-7} \\times 10^{10}\\right) && \\text{Commutative and associative properties of multiplication} \\\\ & =\\left(52.91\\right)\\left(10^{3}\\right) && \\text{Product rule of exponents} \\\\ & =5.291 \\times 10^{4} && \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align} \\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)& = \\left(\\frac{4}{-1.52}\\right)\\left(\\frac{{10}^{5}}{{10}^{9}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(-2.63\\right)\\left({10}^{-4}\\right)&& \\text{Quotient rule of exponents} \\\\ & = -2.63\\times {10}^{-4}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p>3.<br \/>\n[latex]\\begin{align} \\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)& = \\left(2.7\\times 6.04\\right)\\left({10}^{5}\\times {10}^{13}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(16.308\\right)\\left({10}^{18}\\right)&& \\text{Product rule of exponents} \\\\ & = 1.6308\\times {10}^{19}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p>4.<br \/>\n[latex]\\begin{align} \\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)& = \\left(\\frac{1.2}{9.6}\\right)\\left(\\frac{{10}^{8}}{{10}^{5}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(0.125\\right)\\left({10}^{3}\\right)&& \\text{Quotient rule of exponents} \\\\ & = 1.25\\times {10}^{2}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p>5.<br \/>\n[latex]\\begin{align} \\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)& = \\left[3.33\\times \\left(-1.05\\right)\\times 5.62\\right]\\left({10}^{4}\\times {10}^{7}\\times {10}^{5}\\right) \\\\ & \\approx \\left(-19.65\\right)\\left({10}^{16}\\right) \\\\ & = -1.965\\times {10}^{17} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see more examples of writing numbers in scientific notation.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-6\" title=\"Examples:  Write a Number in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fsNu3AdIgdk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Perform the operations and write the answer in scientific notation.<\/p>\n<ol>\n<li>[latex]\\left(-7.5\\times {10}^{8}\\right)\\left(1.13\\times {10}^{-2}\\right)[\/latex]<\/li>\n<li>[latex]\\left(1.24\\times {10}^{11}\\right)\\div \\left(1.55\\times {10}^{18}\\right)[\/latex]<\/li>\n<li>[latex]\\left(3.72\\times {10}^{9}\\right)\\left(8\\times {10}^{3}\\right)[\/latex]<\/li>\n<li>[latex]\\left(9.933\\times {10}^{23}\\right)\\div \\left(-2.31\\times {10}^{17}\\right)[\/latex]<\/li>\n<li>[latex]\\left(-6.04\\times {10}^{9}\\right)\\left(7.3\\times {10}^{2}\\right)\\left(-2.81\\times {10}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q578130\">Show Solution<\/span><\/p>\n<div id=\"q578130\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-8.475\\times {10}^{6}[\/latex]<\/li>\n<li>[latex]8\\times {10}^{-8}[\/latex]<\/li>\n<li>[latex]2.976\\times {10}^{13}[\/latex]<\/li>\n<li>[latex]-4.3\\times {10}^{6}[\/latex]<\/li>\n<li>[latex]\\approx 1.24\\times {10}^{15}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2824&amp;theme=oea&amp;iframe_resize_id=mom3 0\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2828&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2830&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Scientific Notation to Solve Problems<\/h3>\n<p>In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q431376\">Show Solution<\/span><\/p>\n<div id=\"q431376\" class=\"hidden-answer\" style=\"display: none\">\nThe population was [latex]308,000,000=3.08\\times {10}^{8}[\/latex].<\/p>\n<p>The national debt was [latex]\\$ 17,547,000,000,000 \\approx \\$1.75 \\times 10^{13}[\/latex].<\/p>\n<p>To find the amount of debt per citizen, divide the national debt by the number of citizens.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} \\left(1.75\\times {10}^{13}\\right)\\div \\left(3.08\\times {10}^{8}\\right)& = \\left(\\frac{1.75}{3.08}\\right)\\cdot \\left(\\frac{{10}^{13}}{{10}^{8}}\\right) \\\\ & \\approx 0.57\\times {10}^{5}\\hfill \\\\ & = 5.7\\times {10}^{4} \\end{align}[\/latex]<\/div>\n<p>The debt per citizen at the time was about [latex]\\$5.7\\times {10}^{4}[\/latex], or $57,000.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q985422\">Show Solution<\/span><\/p>\n<div id=\"q985422\" class=\"hidden-answer\" style=\"display: none\">\n<p>Number of cells: [latex]3\\times {10}^{13}[\/latex]; length of a cell: [latex]8\\times {10}^{-6}[\/latex] m; total length: [latex]2.4\\times {10}^{8}[\/latex] m or [latex]240,000,000[\/latex] m.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3295&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=101856&amp;theme=oea&amp;iframe_resize_id=mom60\" width=\"100%\" height=\"650\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=102452&amp;theme=oea&amp;iframe_resize_id=mom70\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong>Rules of Exponents<\/strong><br \/>\nFor nonzero real numbers [latex]a[\/latex] and [latex]b[\/latex] and integers [latex]m[\/latex] and [latex]n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Product rule<\/strong><\/td>\n<td>[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Quotient rule<\/strong><\/td>\n<td>[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power rule<\/strong><\/td>\n<td>[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Zero exponent rule<\/strong><\/td>\n<td>[latex]{a}^{0}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Negative rule<\/strong><\/td>\n<td>[latex]{a}^{-n}=\\dfrac{1}{{a}^{n}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power of a product rule<\/strong><\/td>\n<td>[latex]{\\left(a\\cdot b\\right)}^{n}={a}^{n}\\cdot {b}^{n}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Power of a quotient rule<\/strong><\/td>\n<td>[latex]{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>Products of exponential expressions with the same base can be simplified by adding exponents.<\/li>\n<li>Quotients of exponential expressions with the same base can be simplified by subtracting exponents.<\/li>\n<li>Powers of exponential expressions with the same base can be simplified by multiplying exponents.<\/li>\n<li>An expression with exponent zero is defined as 1.<\/li>\n<li>An expression with a negative exponent is defined as a reciprocal.<\/li>\n<li>The power of a product of factors is the same as the product of the powers of the same factors.<\/li>\n<li>The power of a quotient of factors is the same as the quotient of the powers of the same factors.<\/li>\n<li>The rules for exponential expressions can be combined to simplify more complicated expressions.<\/li>\n<li>Scientific notation uses powers of 10 to simplify very large or very small numbers.<\/li>\n<li>Scientific notation may be used to simplify calculations with very large or very small numbers.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>scientific notation&nbsp;<\/strong>a shorthand notation for writing very large or very small numbers in the form [latex]a\\times {10}^{n}[\/latex] where [latex]1\\le |a|<10[\/latex] and [latex]n[\/latex] is an integer\n\n\n<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1741\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Product Rule for Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P0UVIMy2nuI\">https:\/\/youtu.be\/P0UVIMy2nuI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Quotient Rule for Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xy6WW7y_GcU\">https:\/\/youtu.be\/xy6WW7y_GcU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Using the Power Rule to Simplify Expressions With Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/VjcKU5rA7F8\">https:\/\/youtu.be\/VjcKU5rA7F8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 93370, 93399, 93402, 93393. <strong>Authored by<\/strong>: Michael Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1961, 2874. <strong>Authored by<\/strong>: David Lippman. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Simplify Expressions With Zero Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rpoUg32utlc\">https:\/\/youtu.be\/rpoUg32utlc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Expressions With Negative Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Gssi4dBtAEI\">https:\/\/youtu.be\/Gssi4dBtAEI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Power of a Product. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/p-2UkpJQWpo\">https:\/\/youtu.be\/p-2UkpJQWpo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Power of a Quotient. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BoBe31pRxFM\">https:\/\/youtu.be\/BoBe31pRxFM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 44120, 43231. <strong>Authored by<\/strong>: Brenda Gardner. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 7833, 14060. <strong>Authored by<\/strong>: Tyler Wallace. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 109762, 109765. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 51959, 101856, 102452. <strong>Authored by<\/strong>: Roy Shahbazian. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057.. <strong>Authored by<\/strong>: James Sousa. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 43896. <strong>Authored by<\/strong>: Carla Kulinsky. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 2466. <strong>Authored by<\/strong>: Bryan Jones. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3295. <strong>Authored by<\/strong>: Norm Friehauf. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College 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