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Mathematics LibreTexts

10.2: Use Multiplication Properties of Exponents (Part 1)

  • Page ID
    21768
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    Skills to Develop

    • Simplify expressions with exponents
    • Simplify expressions using the Product Property of Exponents
    • Simplify expressions using the Power Property of Exponents
    • Simplify expressions using the Product to a Power Property
    • Simplify expressions by applying several properties
    • Multiply monomials

    be prepared!

    Before you get started, take this readiness quiz.

    1. Simplify: \(\dfrac{3}{4} \cdot \dfrac{3}{4}\). If you missed the problem, review Example 4.3.7.
    2. Simplify: (−2)(−2)(−2). If you missed the problem, review Example 3.7.6.

    Simplify Expressions with Exponents

    Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 means to multiply four factors of 2, so 24 means 2 • 2 • 2 • 2. This format is known as exponential notation.

    Definition: Exponential Notation

    On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.

    This is read a to the mth power.

    In the expression am, the exponent tells us how many times we use the base a as a factor.

    On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.

    Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

    Example \(\PageIndex{1}\):

    Simplify: (a) 53 (b) 91

    Solution

    (a) 53

    Multiply 3 factors of 5. 5 • 5 • 5
    Simplify. 125

    (b) 91

    Multiply 1 factor of 9. 9

    Exercise \(\PageIndex{1}\):

    Simplify: (a) 43 (b) 111

    Answer a

    64

    Answer b

    11

    Exercise \(\PageIndex{2}\):

    Simplify: (a) 34 (b) 211

    Answer a

    81

    Answer b

    21

    Example \(\PageIndex{2}\):

    Simplify: (a) \(\left(\dfrac{7}{8}\right)^{2}\) (b) (0.74)2

    Solution

    (a) \(\left(\dfrac{7}{8}\right)^{2}\)

    Multiply two factors. $$\left(\dfrac{7}{8}\right) \left(\dfrac{7}{8}\right)$$
    Simplify. $$\dfrac{49}{64}$$

    (b) (0.74)2

    Multiply two factors. (0.74)(0.74)
    Simplify. 0.5476

    Exercise \(\PageIndex{3}\):

    Simplify: (a) \(\left(\dfrac{5}{8}\right)^{2}\) (b) (0.67)2

    Answer a

    \(\frac{25}{64}\)

    Answer b

    0.4489

    Exercise \(\PageIndex{4}\):

    Simplify: (a) \(\left(\dfrac{2}{5}\right)^{3}\) (b) (0.127)2

    Answer a

    \(\frac{8}{125}\)

    Answer b

    0.016129

    Example \(\PageIndex{3}\):

    Simplify: (a) (−3)4 (b) −34

    Solution

    (a) (−3)4

    Multiply four factors of −3. (−3)(−3)(−3)(−3)
    Simplify. 81

    (b) −34

    Multiply two factors. −(3 • 3 • 3 • 3)
    Simplify. −81

    Notice the similarities and differences in parts (a) and (b). Why are the answers different? In part (a) the parentheses tell us to raise the (−3) to the 4th power. In part (b) we raise only the 3 to the 4th power and then find the opposite.

    Exercise \(\PageIndex{5}\):

    Simplify: (a) (−2)4 (b) −24

    Answer a

    16

    Answer b

    -16

    Exercise \(\PageIndex{6}\):

    Simplify: (a) (−8)2 (b) −82

    Answer a

    64

    Answer b

    -64

    Simplify Expressions Using the Product Property of Exponents

    You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll 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.

    First, we will look at an example that leads to the Product Property.

    $$x^{2} \cdot x^{2}$$
    What does this mean? How many factors altogether? CNX_BMath_Figure_10_02_015_img-02.png
    So, we have $$x^{5}$$
    Notice that 5 is the sum of the exponents, 2 and 3. $$x^{2} \cdot x^{3}\; is\; x^{2+3},\; or\; x^{5}$$
    We write: $$\begin{split} &x^{2} \cdot x^{3} \\ &x^{2+3} \\ &x^{5} \end{split}$$

    The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

    Definition: Product Property of Exponents

    If a is a real number and m, n are counting numbers, then

    \[a^{m} \cdot a^{n} = a^{m + n}\]

    To multiply with like bases, add the exponents.

    An example with numbers helps to verify this property.

    \[\begin{split} 2^{2} \cdot 2^{3} &\stackrel{?}{=} 2^{2+3} \\ 4 \cdot 8 &\stackrel{?}{=} 2^{5} \\ 32 &= 32\; \checkmark \end{split}\]

    Example \(\PageIndex{4}\):

    Simplify: x5 • x7.

    Solution

    Use the product property, am • an = am + n. $$x^{\textcolor{red}{5+7}}$$
    Simplify. $$x^{12}$$

    Exercise \(\PageIndex{7}\):

    Simplify: x7 • x8.

    Answer

    x15

    Exercise \(\PageIndex{8}\):

    Simplify: x5 • x11.

    Answer

    x16

    Example \(\PageIndex{5}\):

    Simplify: b4 • b.

    Solution

    Rewrite, b = b1. $$b^{4} \cdot b^{1}$$
    Use the product property, am • an = am + n. $$b^{\textcolor{red}{4+1}}$$
    Simplify. $$b^{5}$$

    Exercise \(\PageIndex{9}\):

    Simplify: p9 • p.

    Answer

    p10

    Exercise \(\PageIndex{10}\):

    Simplify: m • m7.

    Answer

    m8

    Example \(\PageIndex{6}\):

    Simplify: 27 • 29.

    Solution

    Use the product property, am • an = am + n. $$2^{\textcolor{red}{7+9}}$$
    Simplify. $$2^{16}$$

    Exercise \(\PageIndex{11}\):

    Simplify: 6 • 69.

    Answer

    610

    Exercise \(\PageIndex{12}\):

    Simplify: 96 • 99.

    Answer

    915

    Example \(\PageIndex{7}\):

    Simplify: y17 • y23.

    Solution

    Notice, the bases are the same, so add the exponents. $$y^{\textcolor{red}{17+23}}$$
    Simplify. $$y^{40}$$

    Exercise \(\PageIndex{13}\):

    Simplify: y24 • y19.

    Answer

    y43

    Exercise \(\PageIndex{14}\):

    Simplify: z15 • z24.

    Answer

    z39

    We can extend the Product Property of Exponents to more than two factors.

    Example \(\PageIndex{8}\):

    Simplify: x3 • x4 • x2.

    Solution

    Add the exponents, since the bases are the same. $$x^{\textcolor{red}{3+4+2}}$$
    Simplify. $$x^{9}$$

    Exercise \(\PageIndex{15}\):

    Simplify: x7 • x5 • x9.

    Answer

    x21

    Exercise \(\PageIndex{16}\):

    Simplify: y3 • y8 • y4.

    Answer

    y15

    Simplify Expressions Using the Power Property of Exponents

    Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

    $$(x^{2})^{3}$$
    What does this mean? $$x^{2} \cdot x^{2} \cdot x^{2}$$
    How many factors altogether? CNX_BMath_Figure_10_02_021_img-03.png
    So, we have $$x^{6}$$
    Notice that 6 is the product of the exponents, 2 and 3. $$(x^{2})^{3}\; is\; x^{2 \cdot 3}\; or\; x^{6}$$
    We write: $$\begin{split} &(x^{2})^{3} \\ &x^{2 \cdot 3} \\ &x^{6} \end{split}$$

    We multiplied the exponents. This leads to the Power Property for Exponents.

    Definition: Power Property of Exponents

    If a is a real number and m, n are whole numbers, then

    \[(a^{m})^{n} = a^{m \cdot n}\]

    To raise a power to a power, multiply the exponents.

    An example with numbers helps to verify this property.

    \[\begin{split} (5^{2})^{3} &\stackrel{?}{=} 5^{2 \cdot 3} \\ (25)^{3} &\stackrel{?}{=} 5^{6} \\ 15,625 &= 15,625\; \checkmark \end{split}\]

    Example \(\PageIndex{9}\):

    Simplify: (a) (x5)7 (b) (36)8

    Solution

    (a) (x5)7

    Use the Power Property, (am)n = am • n. $$x^{\textcolor{red}{5 \cdot 7}}$$
    Simplify. $$x^{35}$$

    (b) (36)8

    Use the Power Property, (am)n = am • n. $$3^{\textcolor{red}{6 \cdot 8}}$$
    Simplify. $$x^{48}$$

    Exercise \(\PageIndex{17}\):

    Simplify: (a) (x7)4 (b) (74)8

    Answer a

    x28

    Answer b

    732

    Exercise \(\PageIndex{18}\):

    Simplify: (a) (x6)9 (b) (86)7

    Answer a

    y54

    Answer b

    842

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