{"id":14266,"date":"2018-06-15T19:32:09","date_gmt":"2018-06-15T19:32:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/exponents-and-scientific-notation\/"},"modified":"2021-12-29T16:40:46","modified_gmt":"2021-12-29T16:40:46","slug":"simplifying-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/simplifying-exponents\/","title":{"raw":"Simplifying Expressions with Exponents","rendered":"Simplifying Expressions with Exponents"},"content":{"raw":"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.\r\n\r\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.\r\n

Rules of Exponents<\/h2>\r\nConsider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, x<\/em>, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.\r\n
[latex]\\begin{array}\\text{ }x^{3}\\cdot x^{4}\\hfill&=\\stackrel{\\text{3 factors } \\text{ 4 factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ \\hfill& =\\stackrel{7 \\text{ factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ \\hfill& =x^{7}\\end{array}[\/latex]<\/div>\r\nThe result is that [latex]{x}^{3}\\cdot {x}^{4}={x}^{3+4}={x}^{7}[\/latex].\r\n\r\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 product rule of exponents.<\/em>\r\n
[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\nNow consider an example with real numbers.\r\n
[latex]{2}^{3}\\cdot {2}^{4}={2}^{3+4}={2}^{7}[\/latex]<\/div>\r\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.\r\n
\r\n

A General Note: The Product Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], the product rule of exponents states that\r\n
[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\n<\/div>\r\n
\r\n

Example: Using the Product Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n
    \r\n \t
  1. [latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\r\n \t
  2. [latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\r\n \t
  3. [latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"878162\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"878162\"]\r\nUse the product rule to simplify each expression.\r\n
      \r\n \t
    1. [latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\r\n \t
    2. [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>\r\n \t
    3. [latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\r\n<\/ol>\r\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.\r\n
      [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>\r\nNotice we get the same result by adding the three exponents in one step.\r\n
      [latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\r\n
      [\/hidden-answer]<\/div>\r\n<\/div>\r\n
      \r\n

      Try It<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n
        \r\n \t
      1. [latex]{k}^{6}\\cdot {k}^{9}[\/latex]<\/li>\r\n \t
      2. [latex]{\\left(\\frac{2}{y}\\right)}^{4}\\cdot \\left(\\frac{2}{y}\\right)[\/latex]<\/li>\r\n \t
      3. [latex]{t}^{3}\\cdot {t}^{6}\\cdot {t}^{5}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"562258\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"562258\"]\r\n
          \r\n \t
        1. [latex]{k}^{15}[\/latex]<\/li>\r\n \t
        2. [latex]{\\left(\\frac{2}{y}\\right)}^{5}[\/latex]<\/li>\r\n \t
        3. [latex]{t}^{14}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n