{"id":10827,"date":"2017-06-05T21:15:52","date_gmt":"2017-06-05T21:15:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10827"},"modified":"2024-04-30T21:31:54","modified_gmt":"2024-04-30T21:31:54","slug":"simplifying-variable-expressions-using-exponent-properties","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/simplifying-variable-expressions-using-exponent-properties\/","title":{"raw":"Simplifying Variable Expressions Using Exponent Properties I","rendered":"Simplifying Variable Expressions Using Exponent Properties I"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions using the Product Property of Exponents<\/li>\r\n \t<li>Simplify expressions using the Power Property of Exponents<\/li>\r\n \t<li>Simplify expressions using the Product to a Power Property of Exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe\u2019ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.\u00a0 First, we will look at a few examples that leads to the Product Property.\r\n\r\nFor example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex]. And don\u2019t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.\r\n\r\nWhat happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].\r\n\r\nWhat about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, [latex]8[\/latex] is the sum of the original two exponents. This concept can be generalized in the following way:\r\n<table id=\"eip-id1168468520971\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{x}^{2}\\cdot{x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What does this mean?\r\n\r\nHow many factors altogether?<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224357\/CNX_BMath_Figure_10_02_015_img-02.png\" alt=\"The equation x times x times x times x times x. There is a brace connecting the first two x's that says 2 factors. A brace connecting the last 3 x's says 3 factors. There is a brace connecting all x's that says 5 factors.\" width=\"180\" height=\"94\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>So, we have<\/td>\r\n<td>[latex]{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}={x}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]5[\/latex] is the sum of the exponents, [latex]2[\/latex] and [latex]3[\/latex].<\/td>\r\n<td>[latex]{x}^{2}\\cdot{x}^{3}[\/latex] is [latex]{x}^{2+3}[\/latex], or [latex]{x}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We write:<\/td>\r\n<td>[latex]{x}^{2}\\cdot {x}^{3}[\/latex]\r\n\r\n[latex]{x}^{2+3}[\/latex]\r\n\r\n[latex]{x}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe base stayed the same and we added the exponents. This leads to the Product Property for Exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Property OF Exponents<\/h3>\r\nFor any real number [latex]x[\/latex] and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.\"\r\n\r\n<\/div>\r\nAn example with numbers helps to verify this property.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {2}^{2}\\cdot {2}^{3}&amp; \\stackrel{?}{=}&amp; {2}^{2+3}\\hfill \\\\ \\hfill 4\\cdot 8&amp; \\stackrel{?}{=}&amp; {2}^{5}\\hfill \\\\ \\hfill 32&amp; =&amp; 32\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{x}^{5}\\cdot {x}^{7}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168466105371\" class=\"unnumbered unstyled\" summary=\"The top line says x to the 5th times x to the 7th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{x}^{5}\\cdot {x}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]x^{\\color{red}{5+7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\r\n[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356596\"]The base of both exponents is <i>a<\/i>, so the product rule applies.\r\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146102[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{b}^{4}\\cdot b[\/latex]\r\n[reveal-answer q=\"338898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338898\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466637014\" class=\"unnumbered unstyled\" summary=\"The top line says b to the 4th times b. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{b}^{4}\\cdot b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite, [latex]b={b}^{1}[\/latex].<\/td>\r\n<td>[latex]{b}^{4}\\cdot {b}^{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]b^{\\color{red}{4+1}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{b}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146107[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{2}^{7}\\cdot {2}^{9}[\/latex]\r\n[reveal-answer q=\"971008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971008\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469803212\" class=\"unnumbered unstyled\" summary=\"The top line says 2 to the 7th times 2 to the 9th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{2}^{7}\\cdot {2}^{9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]2^{\\color{red}{7+9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{16}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146143[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{y}^{17}\\cdot {y}^{23}[\/latex]\r\n[reveal-answer q=\"366560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"366560\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469684383\" class=\"unnumbered unstyled\" summary=\"The top line shows y to the 17 times y to the 23rd. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{y}^{17}\\cdot {y}^{23}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice, the bases are the same, so add the exponents.<\/td>\r\n<td>[latex]y^{\\color{red}{17+23}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{y}^{40}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146144[\/ohm_question]\r\n\r\n<\/div>\r\nWe can extend the Product Property of Exponents to more than two factors.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{x}^{3}\\cdot {x}^{4}\\cdot {x}^{2}[\/latex]\r\n[reveal-answer q=\"278257\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278257\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468510734\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 3rd times x to the 4th times x squared. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{x}^{3}\\cdot {x}^{4}\\cdot {x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the exponents, since the bases are the same.<\/td>\r\n<td>[latex]x^{\\color{red}{3+4+2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146145[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to use the product rule for exponents to simplify expressions.\r\n\r\nhttps:\/\/youtu.be\/P0UVIMy2nuI\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><img class=\"alignleft wp-image-2132\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"Caution\" width=\"96\" height=\"83\" \/><\/h3>\r\nCaution! Do not try to apply this rule to sums.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThink about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]\r\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\r\nNo, it does not because of the order of operations!\r\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\r\nTherefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/hA9AT7QsXWo\r\n<h3>Simplify Expressions Using the Power Property of Exponents<\/h3>\r\nWe will now further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.\u00a0\u00a0See if you can discover a general property.\r\n\r\nLet\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).\r\n\r\n[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].\r\n\r\nSo, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).\r\n\r\nLikewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]\r\n\r\nThis leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].\r\n<table id=\"eip-id1168466113470\" class=\"unnumbered unstyled\" summary=\"The top line shows x squared in parentheses raised to the third power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]({x}^{2})^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{x}^{2}\\cdot{x}^{2}\\cdot{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What does this mean?\r\n\r\nHow many factors altogether?<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224409\/CNX_BMath_Figure_10_02_021_img-03.png\" alt=\"The equation x times x times x times x times x times x. For each sequential pair of x's, there are braces connecting the two x's that say 2 factors. There is a brace connecting all x's that says 6 factors.\" width=\"284\" height=\"95\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>So, we have<\/td>\r\n<td>[latex]{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}={x}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]6[\/latex] is the product of the exponents, [latex]2[\/latex] and [latex]3[\/latex].<\/td>\r\n<td>[latex]({x}^{2})^{3}[\/latex] is [latex]{x}^{2\\cdot3}[\/latex] or [latex]{x}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We write:<\/td>\r\n<td>[latex]{\\left({x}^{2}\\right)}^{3}[\/latex]\r\n\r\n[latex]{x}^{2\\cdot 3}[\/latex]\r\n\r\n[latex]{x}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis leads to the Power Property for Exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>Power Property of Exponents<\/h3>\r\nIf [latex]x[\/latex] is a real number and [latex]a,b[\/latex] are whole numbers, then\r\n\r\n[latex]{\\left({x}^{a}\\right)}^{b}={x}^{a\\cdot b}[\/latex]\r\nTo raise a power to a power, multiply the exponents.\r\n\r\nTake a moment to contrast how this is different from the product rule for exponents found on the previous page.\r\n\r\n<\/div>\r\nAn example with numbers helps to verify this property.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({5}^{2}\\right)}^{3}&amp; \\stackrel{?}{=}&amp; {5}^{2\\cdot 3}\\hfill \\\\ \\hfill {\\left(25\\right)}^{3}&amp; \\stackrel{?}{=}&amp; {5}^{6}\\hfill \\\\ \\hfill 15,625&amp; =&amp; 15,625\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left({x}^{5}\\right)}^{7}[\/latex]\r\n2. [latex]{\\left({3}^{6}\\right)}^{8}[\/latex]\r\n[reveal-answer q=\"411160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411160\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468674718\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 5th in parentheses raised to the 7th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left({x}^{5}\\right)}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td>[latex]x^{\\color{red}{5\\cdot{7}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{35}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047388004\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 to the 6th in parentheses raised to the 8th power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left({3}^{6}\\right)}^{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td>[latex]3^{\\color{red}{6\\cdot{8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{3}^{48}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146148[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].\r\n\r\n[reveal-answer q=\"841688\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841688\"]Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.\r\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]6\\left(c^{4\\cdot 2}\\right)=6c^{8}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWatch the following video to see more examples of how to use the power rule for exponents to simplify expressions.\r\n\r\nhttps:\/\/youtu.be\/Hgu9HKDHTUA\r\n<h3>Simplify Expressions Using the Product to a Power Property<\/h3>\r\nWe will now look at an expression containing a product that is raised to a power. Look for a pattern.\r\n\r\nSimplify this expression.\r\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\r\nNotice that the exponent is applied to each factor of [latex]2a[\/latex]. So, we can eliminate the middle steps.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\r\nThe product of two or more numbers raised to a power is equal to the product of each number raised to the same power.\r\n<table id=\"eip-id1168468541414\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(2x\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What does this mean?<\/td>\r\n<td>[latex]2x\\cdot 2x\\cdot 2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We group the like factors together.<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 2\\cdot x\\cdot x\\cdot x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many factors of [latex]2[\/latex] and of [latex]x?[\/latex]<\/td>\r\n<td>[latex]{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that each factor was raised to the power.<\/td>\r\n<td>[latex]{\\left(2x\\right)}^{3}\\text{ is }{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We write:<\/td>\r\n<td>[latex]{\\left(2x\\right)}^{3}[\/latex]\r\n\r\n[latex]{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>Product to a Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] is a whole number, then\r\n\r\n[latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex]\r\nTo raise a product to a power, raise each factor to that power.\r\n\r\nHow is this rule different from the power raised to a power rule? How is it different from the product rule for exponents shown above?\r\n\r\n<\/div>\r\nAn example with numbers helps to verify this property:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(2\\cdot 3\\right)}^{2}&amp; \\stackrel{?}{=}&amp; {2}^{2}\\cdot {3}^{2}\\hfill \\\\ \\hfill {6}^{2}&amp; \\stackrel{?}{=}&amp; 4\\cdot 9\\hfill \\\\ \\hfill 36&amp; =&amp; 36\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(-11x\\right)}^{2}[\/latex]\r\n[reveal-answer q=\"390160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390160\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466596049\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 11x in parentheses, squared. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-11x\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power of a Product Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\r\n<td>[latex](-11)^{\\color{red}{2}}x^{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]121{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146152[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(3xy\\right)}^{3}[\/latex]\r\n[reveal-answer q=\"678819\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"678819\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466034982\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 3xy to the third. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(3xy\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise each factor to the third power.<\/td>\r\n<td>[latex]3^{\\color{red}{3}}x^{\\color{red}{3}}y^{\\color{red}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]27{x}^{3}{y}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146154[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]\r\n\r\n[reveal-answer q=\"368657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368657\"]\r\n\r\nApply the exponent to each number in the product.\r\n\r\n[latex]2^{6}y^{6}z^{6}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf the variable has an exponent with it, use the Power Rule: multiply the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]\r\n\r\n[reveal-answer q=\"136794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"136794\"]Apply the exponent 2 to each factor within the parentheses.\r\n\r\n[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]\r\n\r\nSquare the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the next video we show more examples of how to simplify a product raised to a power.\r\n\r\nhttps:\/\/youtu.be\/D05D-YIPr1Q\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions using the Product Property of Exponents<\/li>\n<li>Simplify expressions using the Power Property of Exponents<\/li>\n<li>Simplify expressions using the Product to a Power Property of Exponents<\/li>\n<\/ul>\n<\/div>\n<p>We\u2019ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.\u00a0 First, we will look at a few examples that leads to the Product Property.<\/p>\n<p>For example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex]. And don\u2019t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.<\/p>\n<p>What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].<\/p>\n<p>What about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, [latex]8[\/latex] is the sum of the original two exponents. This concept can be generalized in the following way:<\/p>\n<table id=\"eip-id1168468520971\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{x}^{2}\\cdot{x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What does this mean?<\/p>\n<p>How many factors altogether?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224357\/CNX_BMath_Figure_10_02_015_img-02.png\" alt=\"The equation x times x times x times x times x. There is a brace connecting the first two x's that says 2 factors. A brace connecting the last 3 x's says 3 factors. There is a brace connecting all x's that says 5 factors.\" width=\"180\" height=\"94\" \/><\/td>\n<\/tr>\n<tr>\n<td>So, we have<\/td>\n<td>[latex]{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}={x}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]5[\/latex] is the sum of the exponents, [latex]2[\/latex] and [latex]3[\/latex].<\/td>\n<td>[latex]{x}^{2}\\cdot{x}^{3}[\/latex] is [latex]{x}^{2+3}[\/latex], or [latex]{x}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We write:<\/td>\n<td>[latex]{x}^{2}\\cdot {x}^{3}[\/latex]<\/p>\n<p>[latex]{x}^{2+3}[\/latex]<\/p>\n<p>[latex]{x}^{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product Property OF Exponents<\/h3>\n<p>For any real number [latex]x[\/latex] and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/p>\n<p>To multiply exponential terms with the same base, add the exponents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.&#8221;<\/p>\n<\/div>\n<p>An example with numbers helps to verify this property.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {2}^{2}\\cdot {2}^{3}& \\stackrel{?}{=}& {2}^{2+3}\\hfill \\\\ \\hfill 4\\cdot 8& \\stackrel{?}{=}& {2}^{5}\\hfill \\\\ \\hfill 32& =& 32\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{x}^{5}\\cdot {x}^{7}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466105371\" class=\"unnumbered unstyled\" summary=\"The top line says x to the 5th times x to the 7th. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{x}^{5}\\cdot {x}^{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]x^{\\color{red}{5+7}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q356596\">Show Solution<\/span><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">The base of both exponents is <i>a<\/i>, so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146102\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146102&theme=oea&iframe_resize_id=ohm146102&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{b}^{4}\\cdot b[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338898\">Show Solution<\/span><\/p>\n<div id=\"q338898\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466637014\" class=\"unnumbered unstyled\" summary=\"The top line says b to the 4th times b. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{b}^{4}\\cdot b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite, [latex]b={b}^{1}[\/latex].<\/td>\n<td>[latex]{b}^{4}\\cdot {b}^{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]b^{\\color{red}{4+1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{b}^{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146107\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146107&theme=oea&iframe_resize_id=ohm146107&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{2}^{7}\\cdot {2}^{9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971008\">Show Solution<\/span><\/p>\n<div id=\"q971008\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469803212\" class=\"unnumbered unstyled\" summary=\"The top line says 2 to the 7th times 2 to the 9th. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{2}^{7}\\cdot {2}^{9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]2^{\\color{red}{7+9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{16}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146143\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146143&theme=oea&iframe_resize_id=ohm146143&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{y}^{17}\\cdot {y}^{23}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366560\">Show Solution<\/span><\/p>\n<div id=\"q366560\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469684383\" class=\"unnumbered unstyled\" summary=\"The top line shows y to the 17 times y to the 23rd. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{y}^{17}\\cdot {y}^{23}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice, the bases are the same, so add the exponents.<\/td>\n<td>[latex]y^{\\color{red}{17+23}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{y}^{40}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146144\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146144&theme=oea&iframe_resize_id=ohm146144&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We can extend the Product Property of Exponents to more than two factors.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{x}^{3}\\cdot {x}^{4}\\cdot {x}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278257\">Show Solution<\/span><\/p>\n<div id=\"q278257\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468510734\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 3rd times x to the 4th times x squared. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{x}^{3}\\cdot {x}^{4}\\cdot {x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the exponents, since the bases are the same.<\/td>\n<td>[latex]x^{\\color{red}{3+4+2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146145\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146145&theme=oea&iframe_resize_id=ohm146145&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to use the product rule for exponents to simplify expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions Using the Product Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P0UVIMy2nuI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-2132\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"Caution\" width=\"96\" height=\"83\" \/><\/h3>\n<p>Caution! Do not try to apply this rule to sums.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Think about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\n<p>No, it does not because of the order of operations!<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\n<p>Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Simplify Exponential Expressions Using the Product Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hA9AT7QsXWo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Simplify Expressions Using the Power Property of Exponents<\/h3>\n<p>We will now further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.\u00a0\u00a0See if you can discover a general property.<\/p>\n<p>Let\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).<\/p>\n<p>[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].<\/p>\n<p>So, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).<\/p>\n<p>Likewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]<\/p>\n<p>This leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].<\/p>\n<table id=\"eip-id1168466113470\" class=\"unnumbered unstyled\" summary=\"The top line shows x squared in parentheses raised to the third power. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]({x}^{2})^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{x}^{2}\\cdot{x}^{2}\\cdot{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What does this mean?<\/p>\n<p>How many factors altogether?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224409\/CNX_BMath_Figure_10_02_021_img-03.png\" alt=\"The equation x times x times x times x times x times x. For each sequential pair of x's, there are braces connecting the two x's that say 2 factors. There is a brace connecting all x's that says 6 factors.\" width=\"284\" height=\"95\" \/><\/td>\n<\/tr>\n<tr>\n<td>So, we have<\/td>\n<td>[latex]{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}={x}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]6[\/latex] is the product of the exponents, [latex]2[\/latex] and [latex]3[\/latex].<\/td>\n<td>[latex]({x}^{2})^{3}[\/latex] is [latex]{x}^{2\\cdot3}[\/latex] or [latex]{x}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We write:<\/td>\n<td>[latex]{\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n<p>[latex]{x}^{2\\cdot 3}[\/latex]<\/p>\n<p>[latex]{x}^{6}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This leads to the Power Property for Exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>Power Property of Exponents<\/h3>\n<p>If [latex]x[\/latex] is a real number and [latex]a,b[\/latex] are whole numbers, then<\/p>\n<p>[latex]{\\left({x}^{a}\\right)}^{b}={x}^{a\\cdot b}[\/latex]<br \/>\nTo raise a power to a power, multiply the exponents.<\/p>\n<p>Take a moment to contrast how this is different from the product rule for exponents found on the previous page.<\/p>\n<\/div>\n<p>An example with numbers helps to verify this property.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({5}^{2}\\right)}^{3}& \\stackrel{?}{=}& {5}^{2\\cdot 3}\\hfill \\\\ \\hfill {\\left(25\\right)}^{3}& \\stackrel{?}{=}& {5}^{6}\\hfill \\\\ \\hfill 15,625& =& 15,625\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left({x}^{5}\\right)}^{7}[\/latex]<br \/>\n2. [latex]{\\left({3}^{6}\\right)}^{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411160\">Show Solution<\/span><\/p>\n<div id=\"q411160\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468674718\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 5th in parentheses raised to the 7th. The next line says,\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\left({x}^{5}\\right)}^{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td>[latex]x^{\\color{red}{5\\cdot{7}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{35}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047388004\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 to the 6th in parentheses raised to the 8th power. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\left({3}^{6}\\right)}^{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td>[latex]3^{\\color{red}{6\\cdot{8}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{3}^{48}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146148\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146148&theme=oea&iframe_resize_id=ohm146148&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841688\">Show Solution<\/span><\/p>\n<div id=\"q841688\" class=\"hidden-answer\" style=\"display: none\">Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6\\left(c^{4\\cdot 2}\\right)=6c^{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Simplify Exponential Expressions Using the Power Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hgu9HKDHTUA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Simplify Expressions Using the Product to a Power Property<\/h3>\n<p>We will now look at an expression containing a product that is raised to a power. Look for a pattern.<\/p>\n<p>Simplify this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\n<p>Notice that the exponent is applied to each factor of [latex]2a[\/latex]. So, we can eliminate the middle steps.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\n<p>The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.<\/p>\n<table id=\"eip-id1168468541414\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(2x\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What does this mean?<\/td>\n<td>[latex]2x\\cdot 2x\\cdot 2x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We group the like factors together.<\/td>\n<td>[latex]2\\cdot 2\\cdot 2\\cdot x\\cdot x\\cdot x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>How many factors of [latex]2[\/latex] and of [latex]x?[\/latex]<\/td>\n<td>[latex]{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that each factor was raised to the power.<\/td>\n<td>[latex]{\\left(2x\\right)}^{3}\\text{ is }{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We write:<\/td>\n<td>[latex]{\\left(2x\\right)}^{3}[\/latex]<\/p>\n<p>[latex]{2}^{3}\\cdot {x}^{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>Product to a Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] is a whole number, then<\/p>\n<p>[latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex]<br \/>\nTo raise a product to a power, raise each factor to that power.<\/p>\n<p>How is this rule different from the power raised to a power rule? How is it different from the product rule for exponents shown above?<\/p>\n<\/div>\n<p>An example with numbers helps to verify this property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(2\\cdot 3\\right)}^{2}& \\stackrel{?}{=}& {2}^{2}\\cdot {3}^{2}\\hfill \\\\ \\hfill {6}^{2}& \\stackrel{?}{=}& 4\\cdot 9\\hfill \\\\ \\hfill 36& =& 36\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(-11x\\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q390160\">Show Solution<\/span><\/p>\n<div id=\"q390160\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466596049\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 11x in parentheses, squared. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-11x\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power of a Product Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\n<td>[latex](-11)^{\\color{red}{2}}x^{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]121{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146152\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146152&theme=oea&iframe_resize_id=ohm146152&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(3xy\\right)}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q678819\">Show Solution<\/span><\/p>\n<div id=\"q678819\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466034982\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 3xy to the third. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(3xy\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise each factor to the third power.<\/td>\n<td>[latex]3^{\\color{red}{3}}x^{\\color{red}{3}}y^{\\color{red}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]27{x}^{3}{y}^{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146154\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146154&theme=oea&iframe_resize_id=ohm146154&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368657\">Show Solution<\/span><\/p>\n<div id=\"q368657\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the exponent to each number in the product.<\/p>\n<p>[latex]2^{6}y^{6}z^{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If the variable has an exponent with it, use the Power Rule: multiply the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q136794\">Show Solution<\/span><\/p>\n<div id=\"q136794\" class=\"hidden-answer\" style=\"display: none\">Apply the exponent 2 to each factor within the parentheses.<\/p>\n<p>[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]<\/p>\n<p>Square the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the next video we show more examples of how to simplify a product raised to a power.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Simplify Exponential Expressions Using Power Property - Products to Powers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/D05D-YIPr1Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-10827\">\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>Question ID 146154, 146153, 146152. <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><\/li><li>Ex: Simplify Exponential Expressions Using the Power Property of Exponents. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hgu9HKDHTUA\">https:\/\/youtu.be\/Hgu9HKDHTUA<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Simplify Expressions Using the Product Rule of Exponents (Basic). <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>Ex: Simplify Exponential Expressions Using Power Property - Products to Powers. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/D05D-YIPr1Q\">https:\/\/youtu.be\/D05D-YIPr1Q<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":21046,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146154, 146153, 146152\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplify Expressions Using the Product Rule of Exponents (Basic)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/P0UVIMy2nuI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Simplify Exponential Expressions Using the Power Property of Exponents\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Hgu9HKDHTUA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Simplify Exponential Expressions Using Power Property - 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