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Mathematics LibreTexts

6.2: Use Multiplication Properties of Exponents

  • Page ID
    30422
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    Skills to Develop

    By the end of this section, you will be able to:

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

    Note

    Before you get started, take this readiness quiz.

    1. Simplify: \(\frac{3}{4}\cdot \frac{3}{4}\)
      If you missed this problem, review [link].
    2. Simplify: \((−2)(−2)(−2)\).
      If you missed this problem, review [link].

    Simplify Expressions with Exponents

    Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2424 means to multiply 2 by itself 4 times, so 2424 means 2·2·2·22·2·2·2.

    Let’s review the vocabulary for expressions with exponents.

    EXPONENTIAL NOTATION

    This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

    This is read aa to the \(m^{th}\) power.

    In the expression \(a^{m}\), the exponent mm tells us how many times we use the base a as a factor.

    This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

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

    Exercise \(\PageIndex{1}\)

    Simplify:

    1. \(4^{3}\)
    2. \(7^{1}\)
    3. \(\left(\frac{5}{6}\right)^{2}\)
    4. \((0.63)^{2}\)
    Answer
    1. \(\begin{array}{ll} & 4^{3}\\ {\text { Multiply three factors of } 4 .} & {4 \cdot 4 \cdot 4} \\ {\text { Simplify. }} & {64}\end{array}\)
    2. \(\begin{array}{ll} & 7^{1}\\ \text{Multiply one factor of 7.} & 7\end{array}\)
    3. \(\begin{array}{ll} &\left(\frac{5}{6}\right)^{2}\\ {\text { Multiply two factors. }} & {\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)} \\ {\text { Simplify. }} & {\frac{25}{36}}\end{array}\)
    4. \(\begin{array}{ll} &(0.63)^{2}\\ {\text { Multiply two factors. }} & {(0.63)(0.63)} \\ {\text { Simplify. }} & {0.3969}\end{array}\)

    Exercise \(\PageIndex{2}\)

    Simplify:

    1. \(6^{3}\)
    2. \(15^{1}\)
    3. \(\left(\frac{3}{7}\right)^{2}\)
    4. \((0.43)^{2}\)
    Answer
    1. 216
    2. 15
    3. \(\frac{9}{49}\)
    4. 0.1849

    Exercise \(\PageIndex{3}\)

    Simplify:

    1. \(2^{5}\)
    2. \(21^{1}\)
    3. \(\left(\frac{2}{5}\right)^{3}\)
    4. \((0.218)^{2}\)
    Answer
    1. 32
    2. 21
    3. \(\frac{8}{125}\)
    4. 0.047524

    Exercise \(\PageIndex{4}\)

    Simplify:

    1. \((-5)^{4}\)
    2. \(-5^{4}\)
    Answer
    1. \(\begin{array}{ll} &(-5)^{4}\\{\text { Multiply four factors of }-5} & {(-5)(-5)(-5)} \\ {\text { Simplify. }} & {625}\end{array}\)
    2. \(\begin{array}{ll} &-5^{4}\\{\text { Multiply four factors of } 5 .} & {-(5 \cdot 5 \cdot 5 \cdot 5)} \\ {\text { Simplify. }} & {-625}\end{array}\)

    Exercise \(\PageIndex{5}\)

    Simplify:

    1. \((-3)^{4}\)
    2. \(-3^{4}\)
    Answer
    1. 81
    2. −81

    Exercise \(\PageIndex{6}\)

    Simplify:

    1. \((-13)^{4}\)
    2. \(-13^{4}\)
    Answer
    1. 169
    2. −169

    Notice the similarities and differences in Exercise \(\PageIndex{4}\)1 and Exercise \(\PageIndex{4}\)2! Why are the answers different? As we follow the order of operations in part 1 the parentheses tell us to raise the (−5) to the 4th power. In part 2 we raise just the 5 to the 4th power and then take the opposite.

    Simplify Expressions Using the Product Property for 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.

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

    x squared times x cubed.
    What does this mean?
    How many factors altogether?
    x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.
    So, we have x to the fifth power.
    Notice that 5 is the sum of the exponents, 2 and 3. x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.

    We write: \[\begin{array}{c}{x^{2} \cdot x^{3}} \\ {x^{2+3}} \\ {x^{5}}\end{array}\]

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

    PRODUCT PROPERTY FOR EXPONENTS

    If a is a real number, and m and 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.

    \[\begin{array}{rll} {2^3\cdot2^2} &\stackrel{?}{=} & 2^{2+3}\\ {4\cdot 8} &\stackrel{?}{=} & 2^{5} \\ {32} &=& 32\checkmark\end{array}\]

    Exercise \(\PageIndex{7}\)

    Simplify: \(y^{5} \cdot y^{6}\)

    Answer
    y to the fifth power times y to the sixth power.
    Use the product property, \(a^{m} \cdot a^{n}=a^{m+n}\). y to the power of 5 plus 6.
    Simplify. y to the eleventh power.

    Exercise \(\PageIndex{8}\)

    Simplify: \(b^{9} \cdot b^{8}\)

    Answer

    \(b^{17}\)

    Exercise \(\PageIndex{9}\)

    Simplify: \(x^{12} \cdot x^{4}\)

    Answer

    \(x^{16}\)

    Exercise \(\PageIndex{10}\)

    Simplify:

    1. \(2^{5} \cdot 2^{9}\)
    2. \(3\cdot 3^{4}\)
    Answer

    1.

    2 to the fifth power times 2 to the ninth power.
    Use the product property, \(a^{m} \cdot a^{n}=a^{m+n}\). 2 to the power of 5 plus 9.
    Simplify. 2 to the 14th power.

    2.

    3 to the fifth power times 3 to the fourth power.
    Use the product property, \(a^{m} \cdot a^{n}=a^{m+n}\). 3 to the power of 5 plus 4.
    Simplify. 3 to the ninth power.

    Exercise \(\PageIndex{11}\)

    Simplify:

    1. \(5\cdot 5^{5}\)
    2. \(4^{9} \cdot 4^{9}\)
    Answer
    1. \(5^{6}\)
    2. \(4^{18}\)

    Exercise \(\PageIndex{12}\)

    Simplify:

    1. \(7^{6} \cdot 7^{8}\)
    2. \(10 \cdot 10^{10}\)
    Answer
    1. \(7^{14}\)
    2. \(10^{11}\)

    Exercise \(\PageIndex{13}\)

    Simplify:

    1. \(a^{7} \cdot a\)
    2. \(x^{27} \cdot x^{13}\)
    Answer

    1.

    a to the seventh power times a.
    Rewrite, a = a1. a to the seventh power times a to the first power.
    Use the product property, am · an = am+n. a to the power of 7 plus 1.
    Simplify. a to the eighth power.

    2.

    x to the twenty-seventh power times x to the thirteenth power.
    Notice, the bases are the same, so add the exponents. x to the power of 27 plus 13.
    Simplify. x to the fortieth power.

    Exercise \(\PageIndex{14}\)

    Simplify:

    1. \(p^{5} \cdot p\)
    2. \(y^{14} \cdot y^{29}\)
    Answer
    1. \(p^{6}\)
    2. \(y^{43}\)

    Exercise \(\PageIndex{15}\)

    Simplify:

    1. \(z \cdot z^{7}\)
    2. \(b^{15} \cdot b^{34}\)
    Answer
    1. \(z^{8}\)
    2. \(b^{49}\)

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

    Exercise \(\PageIndex{16}\)

    Simplify: \(d^{4} \cdot d^{5} \cdot d^{2}\)

    Answer
    d to the fourth power times d to the fifth power times d squared.
    Add the exponents, since bases are the same. d to the power of 4 plus 5 plus 2.
    Simplify. d to the eleventh power.

    Exercise \(\PageIndex{17}\)

    Simplify: \(x^{6} \cdot x^{4} \cdot x^{8}\)

    Answer

    \(x^{18}\)

    Exercise \(\PageIndex{18}\)

    Simplify: \(b^{5} \cdot b^{9} \cdot b^{5}\)

    Answer

    \(b^{19}\)

    Simplify Expressions Using the Power Property for 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 squared, in parentheses, cubed.
    What does this mean?
    How many factors altogether?
    x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.
    So we have x to the sixth power.
    Notice that 6 is the product of the exponents, 2 and 3. x squared cubed is x to the power of 2 times 3, or x to the sixth power.

    We write:

    \[\begin{array}{c}{\left(x^{2}\right)^{3}} \\ {x^{2 \cdot 3}} \\ {x^{6}}\end{array}\]

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

    POWER PROPERTY FOR EXPONENTS

    If aa is a real number, and mandnmandn are whole numbers, then

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

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

    An example with numbers helps to verify this property.

    \[\begin{array} {lll} \left(3^{2}\right)^{3} &\stackrel{?}{=}&3^{2 \cdot 3} \\(9)^{3} &\stackrel{?}{=} & 3^{6} \\ 729 &=&729\checkmark \end{array}\]

    Exercise \(\PageIndex{19}\)

    Simplify:

    1. \(\left(y^{5}\right)^{9}\)
    2. \(\left(4^{4}\right)^{7}\)
    Answer

    1.

    y to the fifth power, in parentheses, to the ninth power.
    Use the power property, (am)n = am·n. y to the power of 5 times 9.
    Simplify. y to the 45th power.

    2.

    4 to the fourth power, in parentheses, to the 7th power.
    Use the power property. 4 to the power of 4 times 7.
    Simplify. 4 to the twenty-eighth power.

    Exercise \(\PageIndex{20}\)

    Simplify:

    1. \( \left(b^{7}\right)^{5} \)
    2. \(\left(5^{4}\right)^{3}\)
    Answer
    1. \( b^{35}\)
    2. \(5^{12}\)

    Exercise \(\PageIndex{21}\)

    Simplify:

    1. \(\left(z^{6}\right)^{9}\)
    2. \(\left(3^{7}\right)^{7}\)
    Answer
    1. \(z^{54}\)
    2. \(3^{49}\)

    Simplify Expressions Using the Product to a Power Property

    We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

    \(\begin{array}{ll}{\text { What does this mean? }} & {\text { (2x) }^{3}} \\ {\text { We group the like factors together. }} & {2 x \cdot 2 x \cdot 2 x} \\ {\text { How many factors of } 2 \text { and of } x ?} & {2 \cdot 2 \cdot x^{3}} \\ {\text { Notice that each factor was raised to the power and }(2 x)^{3} \text { is } 2^{3} \cdot x^{3}}\end{array}\)

    \(\begin{array}{ll}\text{We write:} & {(2 x)^{3}} \\ & {2^{3} \cdot x^{3}}\end{array}\)

    The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

    PRODUCT TO A POWER PROPERTY FOR EXPONENTS

    If a and b are real numbers and mm is a whole number, then

    \[(a b)^{m}=a^{m} b^{m}\]

    To raise a product to a power, raise each factor to that power.

    An example with numbers helps to verify this property:

    \[\begin{array}{lll}(2 \cdot 3)^{2} &\stackrel{?}{=}&2^{2} \cdot 3^{2} \\ 6^{2} &\stackrel{?}{=}&4 \cdot 9 \\ 36 &=&36
    \checkmark \end{array}\]

    Exercise \(\PageIndex{22}\)

    Simplify:

    1. \((-9 d)^{2}\)
    2. \((3mn)^{3}\).
    Answer

    1.

    Negative 9 d squared.
    Use Power of a Product Property, (ab)m = ambm. negative 9 squared d squared.
    Simplify. 81 d squared.
    2.
    3 m n cubed.
    Use Power of a Product Property, (ab)m = ambm. 3 cubed m cubed n cubed.
    Simplify. 27 m cubed n cubed.

    Exercise \(\PageIndex{23}\)

    Simplify:

    1. \((-12 y)^{2}\)
    2. \((2 w x)^{5}\)
    Answer
    1. \(144y^{2}\)
    2. \(32w^{5} x^{5}\)

    Exercise \(\PageIndex{24}\)

    Simplify:

    1. \((5 w x)^{3}\)
    2. \((-3 y)^{3}\)
    Answer
    1. 125\(w^{3} x^{3}\)
    2. \(-27 y^{3}\)

    Simplify Expressions by Applying Several Properties

    We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

    PROPERTIES OF EXPONENTS

    If a and b are real numbers, and m and n are whole numbers, then

    \[\begin{array}{llll} \textbf{Product Property } & a^{m} \cdot a^{n}&=&a^{m+n} \\ \textbf {Power Property } &\left(a^{m}\right)^{n}&=&a^{m n} \\ \textbf {Product to a Power } &(a b)^{m}&=&a^{m} b^{m} \end{array}\]

    All exponent properties hold true for any real numbers m and n. Right now, we only use whole number exponents.

    Exercise \(\PageIndex{25}\)

    Simplify:

    1. \(\left(y^{3}\right)^{6}\left(y^{5}\right)^{4}\)
    2. \(\left(-6 x^{4} y^{5}\right)^{2}\)
    Answer
    1. \(\begin{array}{ll}& \left(y^{3}\right)^{6}\left(y^{5}\right)^{4}\\ {\text { Use the Power Property. }}& y^{18} \cdot y^{20} \\ {\text { Add the exponents. }} & y^{38} \end{array}\)
    2. \(\begin{array}{ll}& \left(-6 x^{4} y^{5}\right)^{2}\\ {\text { Use the Product to a Power Property. }} & {(-6)^{2}\left(x^{4}\right)^{2}\left(y^{5}\right)^{2}} \\ {\text { Use the Power Property. }} & {(-6)^{2}\left(x^{8}\right)\left(y^{10}\right)^{2}} \\ {\text { Simplify. }} & {36 x^{8} y^{10}}\end{array}\)

    Exercise \(\PageIndex{26}\)

    Simplify:

    1. \(\left(a^{4}\right)^{5}\left(a^{7}\right)^{4}\)
    2. \(\left(-2 c^{4} d^{2}\right)^{3}\)
    Answer
    1. \(a^{48}\)
    2. \(-8 c^{12} d^{6}\)

    Exercise \(\PageIndex{27}\)

    Simplify:

    1. \(\left(-3 x^{6} y^{7}\right)^{4}\)
    2. \(\left(q^{4}\right)^{5}\left(q^{3}\right)^{3}\)
    Answer
    1. 81\(x^{24} y^{28}\)
    2. \(q^{29}\)

    Exercise \(\PageIndex{28}\)

    Simplify:

    1. \((5 m)^{2}\left(3 m^{3}\right)\)
    2. \(\left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3}\)
    Answer
    1. \(\begin{array}{ll}& (5 m)^{2}\left(3 m^{3}\right)\\{\text { Raise } 5 m \text { to the second power. }} & {5^{2} m^{2} \cdot 3 m^{3}} \\ {\text { Simplify. }} & {25 m^{2} \cdot 3 m^{3}} \\ {\text { Use the Commutative Property. }} & {25 \cdot 3 \cdot m^{2} \cdot m^{3}} \\ {\text { Multiply the constants and add the exponents. }} & {75 m^{5}}\end{array}\)
    2. \(\begin{array}{ll} & \left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3} \\ \text{Use the Product to a Power Property.} & \left(3^{4} x^{8} y^{4}\right)\left(2^{3} x^{3} y^{6}\right)\\\text{Simplify.} & \left(81 x^{8} y^{4}\right)\left(8 x^{3} y^{6}\right)\\ \text{Use the Commutative Property.} &81\cdot 8 \cdot x^{8} \cdot x^{3} \cdot y^{4} \cdot y^{6} \\\text{Multiply the constants and add the exponents.} & 648x^{11} y^{10}\\ \end{array}\)

    Exercise \(\PageIndex{29}\)

    Simplify:

    1. \((5 n)^{2}\left(3 n^{10}\right)\)
    2. \(\left(c^{4} d^{2}\right)^{5}\left(3 c d^{5}\right)^{4}\)
    Answer
    1. 75\(n^{12}\)
    2. 81\(c^{24} d^{30}\)

    Exercise \(\PageIndex{30}\)

    Simplify:

    1. \(\left(a^{3} b^{2}\right)^{6}\left(4 a b^{3}\right)^{4}\)
    2. \((2 x)^{3}\left(5 x^{7}\right)\)
    Answer
    1. 256\(a^{22} b^{24}\)
    2. 40\(x^{10}\)

    Multiply Monomials

    Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

    Exercise \(\PageIndex{31}\)

    Multiply: \(\left(3 x^{2}\right)\left(-4 x^{3}\right)\)

    Answer

    \(\begin{array}{ll} & \left(3 x^{2}\right)\left(-4 x^{3}\right)\\ \text{Use the Commutative Property to rearrange the terms.} & 3\cdot(-4) \cdot x^{2} \cdot x^{3}\\
    \text{Multiply.} & -12 x^{5}\end{array}\)

    Exercise \(\PageIndex{32}\)

    Multiply: \(\left(5 y^{7}\right)\left(-7 y^{4}\right)\)

    Answer

    \(-35 y^{11}\)

    Exercise \(\PageIndex{33}\)

    Multiply: \(\left(-6 b^{4}\right)\left(-9 b^{5}\right)\)

    Answer

    54\(b^{9}\)

    Exercise \(\PageIndex{34}\)

    Multiply: \(\left(\frac{5}{6} x^{3} y\right)\left(12 x y^{2}\right)\)

    Answer

    \(\begin{array}{ll} & \left(\frac{5}{6} x^{3} y\right)\left(12 x y^{2}\right)\\ \text{Use the Commutative Property to rearrange the terms.} & \frac{5}{6} \cdot 12 \cdot x^{3} \cdot x \cdot y \cdot y^{2}\\ \text{Multiply.} &10x^{4} y^{3}\end{array}\)

    Exercise \(\PageIndex{35}\)

    Multiply: \(\left(\frac{2}{5} a^{4} b^{3}\right)\left(15 a b^{3}\right)\)

    Answer

    6\(a^{5} b^{6}\)

    Exercise \(\PageIndex{36}\)

    Multiply: \(\left(\frac{2}{3} r^{5} s\right)\left(12 r^{6} s^{7}\right)\)

    Answer

    8\(r^{11} s^{8}\)

    Note

    Access these online resources for additional instruction and practice with using multiplication properties of exponents:

    Key Concepts

    • Exponential Notation
      This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m powder means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.
    • Properties of Exponents
      • If a,b are real numbers and m,nm,n are whole numbers, then

    \[\begin{array}{llll} \textbf{Product Property } & a^{m} \cdot a^{n}&=&a^{m+n} \\ \textbf {Power Property } &\left(a^{m}\right)^{n}&=&a^{m n} \\ \textbf {Product to a Power } &(a b)^{m}&=&a^{m} b^{m} \end{array}\]

    Practice Makes Perfect

    Simplify Expressions with Exponents

    In the following exercises, simplify each expression with exponents.

    Exercise \(\PageIndex{37}\)

    1. \(3^5\)
    2. \(9^1\)
    3. \((\frac{1}{3})^2\)
    4. \((0.2)^4\)

    Exercise \(\PageIndex{38}\)

    1. \(10^4\)
    2. \(17^1\)
    3. \((\frac{2}{9})^2\)
    4. \((0.5)^3\)
    Answer
    1. 10,000
    2. 17
    3. \(\frac{4}{81}\)
    4. 0.125

    Exercise \(\PageIndex{39}\)

    1. \(2^6\)
    2. \(14^1\)
    3. \((\frac{2}{5})^3\)
    4. \((0.7)^2\)

    Exercise \(\PageIndex{40}\)

    1. \(8^3\)
    2. \(8^1\)
    3. \((\frac{3}{4})^3\)
    4. \((0.4)^3\)
    Answer
    1. 512
    2. 8
    3. \(\frac{27}{64}\)
    4. 0.064

    Exercise \(\PageIndex{41}\)

    1. \((−6)^4\)
    2. \(−6^4\)

    Exercise \(\PageIndex{42}\)

    1. \((−2)^6\)
    2. \(−2^6\)
    Answer
    1. 64
    2. −64

    Exercise \(\PageIndex{43}\)

    1. \(−(\frac{1}{4})^4\)
    2. \((−\frac{1}{4})^4\)

    Exercise \(\PageIndex{44}\)

    1. \(−(\frac{2}{3})^2\)
    2. \((−\frac{2}{3})^2\)
    Answer
    1. \(−\frac{4}{9}\)
    2. \(\frac{4}{9}\)

    Exercise \(\PageIndex{45}\)

    1. \(−0.5^2\)
    2. \((−0.5)^2\)

    Exercise \(\PageIndex{46}\)

    1. \(−0.1^4\)
    2. \((−0.1)^4\)
    Answer
    1. −0.001
    2. 0.001

    Simplify Expressions Using the Product Property for Exponents

    In the following exercises, simplify each expression using the Product Property for Exponents.

    Exercise \(\PageIndex{47}\)

    \(d^3·d^6\)

    Exercise \(\PageIndex{48}\)

    \(x^4·x^2\)

    Answer

    \(x^6\)

    Exercise \(\PageIndex{49}\)

    \(n^{19}·n^{12}\)

    Exercise \(\PageIndex{50}\)

    \(q^{27}·q^{15}\)

    Answer

    \(q^{42}\)

    Exercise \(\PageIndex{51}\)

    1. \(4^5·4^9\)
    2. \(8^9·8\)

    Exercise \(\PageIndex{52}\)

    1. \(3^{10}·3^6\)
    2. \(5·5^{4}\)
    Answer
    1. \(3^{16}\)
    2. \(5^5\)

    Exercise \(\PageIndex{53}\)

    1. \(y·y^3\)
    2. \(z^{25}·z^8\)

    Exercise \(\PageIndex{54}\)

    1. \(w^5·w\)
    2. \(u^{41}·u^{53}\)
    Answer
    1. \(w^6\)
    2. \(u^{94}\)

    Exercise \(\PageIndex{55}\)

    \(w·w^2·w^3\)

    Exercise \(\PageIndex{56}\)

    \(y·y^3·y^5\)

    Answer

    \(y^9\)

    Exercise \(\PageIndex{57}\)

    \(a^4·a^3·a^9\)

    Exercise \(\PageIndex{58}\)

    \(c^5·c^{11}·c^2\)

    Answer

    \(c^{18}\)

    Exercise \(\PageIndex{59}\)

    \(m^x·m^3\)

    Exercise \(\PageIndex{60}\)

    \(n^y·n^2\)

    Answer

    \(n^{y+2}\)

    Exercise \(\PageIndex{61}\)

    \(y^a·y^b\)

    Exercise \(\PageIndex{62}\)

    \(x^p·x^q\)

    Answer

    \(x^{p+q}\)

    Simplify Expressions Using the Power Property for Exponents

    In the following exercises, simplify each expression using the Power Property for Exponents.

    Exercise \(\PageIndex{63}\)

    1. \((m^4)^2\)
    2. \( (10^3)^6\)

    Exercise \(\PageIndex{64}\)

    1. \((b^2)^7\)
    2. \((3^8)^2\)
    Answer
    1. \(b^{14}\)
    2. \(3^{16}\)

    Exercise \(\PageIndex{65}\)

    1. \((y^3)^x\)
    2. \((5^x)^y\)

    Exercise \(\PageIndex{66}\)

    1. \((x^2)^y\)
    2. \((7^a)^b\)
    Answer
    1. \(x^{2y}\)
    2. \(7^{ab}\)

    Simplify Expressions Using the Product to a Power Property

    In the following exercises, simplify each expression using the Product to a Power Property.

    Exercise \(\PageIndex{67}\)

    1. \((6a)^2\)
    2. \((3xy)^2\)

    Exercise \(\PageIndex{68}\)

    1. \((5x)^2\)
    2. \((4ab)^2\)
    Answer
    1. \(25x^2\)
    2. \(16a^{2}b^{2}\)

    Exercise \(\PageIndex{69}\)

    1. \((−4m)^3\)
    2. \((5ab)^3\)

    Exercise \(\PageIndex{70}\)

    1. \((−7n)^3\)
    2. \((3xyz)^4\)
    Answer
    1. \(−343n^3\)
    2. \(81x^{4}y^{4}z^{4}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{71}\)

    1. \((y^2)^4·(y^3)^2\)
    2. \((10a^{2}b)^3\)

    Exercise \(\PageIndex{72}\)

    1. \((w^4)^3·(w^5)^2\)
    2. \((2xy^4)^5\)
    Answer
    1. \(w^{22}\)
    2. \(32x^{5}y^{20}\)

    Exercise \(\PageIndex{73}\)

    1. \((−2r^{3}s^2)^4\)
    2. \((m^5)^3·(m^9)^4\)

    Exercise \(\PageIndex{74}\)

    1. \((−10q^{2}p^4)^3\)
    2. \((n^3)^{10}·(n^5)^2\)
    Answer
    1. \(−1000q^{6}p^{12}\)
    2. \(n^{40}\)

    Exercise \(\PageIndex{75}\)

    1. \((3x)^{2}(5x)\)
    2. \((5t^2)^{3}(3t)^{2}\)

    Exercise \(\PageIndex{76}\)

    1. \((2y)^{3}(6y)\)
    2. \((10k^4)^{3}(5k^6)^{2}\)
    Answer
    1. \(48y^4\)
    2. \(25,000k^{24}\)

    Exercise \(\PageIndex{77}\)

    1. \((5a)^{2}(2a)^3\)
    2. \((12y^2)^{3}(23y)^2\)

    Exercise \(\PageIndex{78}\)

    1. \((4b)^{2}(3b)^{3}\)
    2. \((12j^2)^{5}(25j^3)^2\)
    Answer
    1. \(432b^5\)
    2. \(1200j^{16}\)

    Exercise \(\PageIndex{79}\)

    1. \((25x^{2}y)^3\)
    2. \((89xy^4)^2\)

    Exercise \(\PageIndex{80}\)

    1. \((2r^2)^{3}(4r)^2\)
    2. \((3x^3)^{3}(x^5)^4\)
    Answer
    1. \(128r^{8}\)
    2. \(1200j^{16}\)

    Exercise \(\PageIndex{81}\)

    1. \((m^{2}n)^{2}(2mn^5)^4\)
    2. \((3pq^4)^{2}(6p^{6}q)^2\)
    ​​​​​​Multiply Monomials

    In the following exercises, multiply the monomials.

    Exercise \(\PageIndex{82}\)

    \((6y^7)(−3y^4)\)

    Answer

    \(−18y^{11}\)

    Exercise \(\PageIndex{83}\)

    \((−10x^5)(−3x^3)\)

    Exercise \(\PageIndex{84}\)

    \((−8u^6)(−9u)\)

    Answer

    \(72u^{7}\)

    Exercise \(\PageIndex{85}\)

    \((−6c^4)(−12c)\)

    Exercise \(\PageIndex{86}\)

    \((\frac{1}{5}f^8)(20f^3)\)

    Answer

    \(4f^{11}\)

    Exercise \(\PageIndex{87}\)

    \((\frac{1}{4}d^5)(36d^2)\)

    Exercise \(\PageIndex{88}\)

    \((4a^{3}b)(9a^{2}b^6)\)

    Answer

    \(36a^{5}b^7\)

    Exercise \(\PageIndex{89}\)

    \((6m^{4}n^3)(7mn^5)\)

    Exercise \(\PageIndex{90}\)

    \((\frac{4}{7}rs^2)(14rs^3)\)

    Answer

    \(8r^{2}s^5\)

    Exercise \(\PageIndex{91}\)

    \((\frac{5}{8}x^{3}y)(24x^{5}y)\)

    Exercise \(\PageIndex{92}\)

    \((\frac{2}{3}x^{2}y)(\frac{3}{4}xy^2)\)

    Answer

    \(\frac{1}{2}x^{3}y^3\)

    Exercise \(\PageIndex{93}\)

    \((\frac{3}{5}m^{3}n^2)(\frac{5}{9}m^{2}n^3)\)

    ​​​​​​Mixed Practice

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{94}\)

    \((x^2)^4·(x^3)^2\)

    Answer

    \(x^{14}\)

    Exercise \(\PageIndex{95}\)

    \((y^4)^3·(y^5)^2\)

    Exercise \(\PageIndex{96}\)

    \((a^2)^6·(a^3)^8\)

    Answer

    \(a^{36}\)

    Exercise \(\PageIndex{97}\)

    \((b^7)^5·(b^2)^6\)

    Exercise \(\PageIndex{98}\)

    \((2m^6)^3\)

    Answer

    \(8m^{18}\)

    Exercise \(\PageIndex{99}\)

    \((3y^2)^4\)

    Exercise \(\PageIndex{100}\)

    \((10x^{2}y)^3\)

    Answer

    \(1000x^{6}y^3\)

    Exercise \(\PageIndex{101}\)

    \((2mn^4)^5\)

    Exercise \(\PageIndex{102}\)

    \((−2a^{3}b^2)^4\)

    Answer

    \(16a^{12}b^8\)

    Exercise \(\PageIndex{103}\)

    \((−10u^{2}v^4)^3\)

    Exercise \(\PageIndex{104}\)

    \((\frac{2}{3}x^{2}y)^3\)

    Answer

    \(\frac{8}{27}x^{6}y^3\)

    Exercise \(\PageIndex{105}\)

    \((\frac{7}{9}pq^4)^2\)

    Exercise \(\PageIndex{107}\)

    \((8a^3)^{2}(2a)^4\)

    Answer

    \(1024a^{10}\)

    Exercise \(\PageIndex{108}\)

    \((5r^2)^{3}(3r)^2\)

    Exercise \(\PageIndex{109}\)

    \((10p^4)^{3}(5p^6)^2\)

    Answer

    \(25000p^{24}\)

    Exercise \(\PageIndex{110}\)

    \((4x^3)^{3}(2x^5)^4\)

    Exercise \(\PageIndex{111}\)

    \((\frac{1}{2}x^{2}y^3)^{4}(4x^{5}y^3)^2\)

    Answer

    \(x^{18}y^{18}\)

    Exercise \(\PageIndex{112}\)

    \((\frac{1}{3}m^{3}n^2)^{4}(9m^{8}n^3)^2\)

    Exercise \(\PageIndex{113}\)

    \((3m^{2}n)^{2}(2mn^5)^4\)

    Answer

    \(144m^{8}n^{22}\)

    Exercise \(\PageIndex{114}\)

    \((2pq^4)^{3}(5p^{6}q)^2\)

    ​​​​​​Everyday Math

    Exercise \(\PageIndex{115}\)

    Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is \(10^2\), on the third round is \(10^3\), as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

    1 10
    2 \(10^2\)
    3 \(10^3\)
    ... ...
    6 ?
    Answer

    1,000,000

    Exercise \(\PageIndex{116}\)

    Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $35,000(1.03), after 2 years was $\(35,000(1.03)^2\), after 3 years was $\(35,000(1.03)^3\), as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

    1 $35,000(1.03)
    2 $\(35,000(1.03)^2\)
    3 $\(35,000(1.03)^3\)
    ... ...
    10 ?

    Exercise \(\PageIndex{117}\)

    Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $1,000(0.70) and the cost of the item during the second week would be $\(1,000(0.70)^2\). Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

    1 $1,000(0.70)
    2 $\(1,000(0.70)^2\)
    3
    ... ...
    8 ?
    Answer

    $168.07

    Exercise \(\PageIndex{118}\)

    Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for $20,000, the value at the end of the first year would be $20,000(0.90) and the value of the car after the end of the second year would be $\(20,000(0.90)^2\). Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

    1 $20,000(0.90)
    2 $\(20,000(0.90)^2\)
    3
    ... ...
    8 ?

    Writing Exercises

    Exercise \(\PageIndex{119}\)

    Use the Product Property for Exponents to explain why \(x·x=x^2\)

    Answer

    Answers will vary.

    Exercise \(\PageIndex{120}\)

    Explain why \(−5^3=(−5)^3\), but \(−5^4 \ne (−5)^4\).

    Exercise \(\PageIndex{121}\)

    Jorge thinks \((\frac{1}{2})^2\) is 1. What is wrong with his reasoning?

    Answer

    Answers will vary.

    Exercise \(\PageIndex{122}\)

    Explain why \(x^3·x^5\) is \(x^8\), and not \(x^{15}\).

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has seven rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions with exponents,” “simplify expressions using the Product Property for Exponents,” “simplify expressions using the Power Property for Exponents,” “simplify expressions using the Product to a Power Property,” “simplify expressions by applying several properties,” and “multiply monomials.” The rest of the cells are blank.

    ⓑ After reviewing this checklist, what will you do to become confident for all goals?

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