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Mathematics LibreTexts

10.3: Use Multiplication Properties of Exponents (Part 2)

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    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. Look for a pattern.

    (2x)3
    What does this mean? 2x • 2x • 2x
    We group the like factors together. 2 • 2 • 2 • x • x • x
    How many factors of 2 and of x? 23 • x3
    Notice that each factor was raised to the power. (2x)3 is 23 • x3
    We write: $$\begin{split} &(2x)^{3} \\ &2^{3} \cdot x^{3} \end{split}$$

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

    Definition: Product to a Power Property of Exponents

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

    \[(ab)^{m} = a^{m} b^{m} \tag{10.2.27}\]

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

    An example with numbers helps to verify this property:

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

    Example \(\PageIndex{10}\):

    Simplify: (−11x)2.

    Solution

    Use the Power of a Product Property, (ab)m = am bm. $$(-11)^{\textcolor{red}{2}} x^{\textcolor{red}{2}} \tag{10.2.28}$$
    Simplify. $$121x^{2} \tag{10.2.29}$$

    Exercise \(\PageIndex{19}\):

    Simplify: (−14x)2.

    Answer

    196x2

    Exercise \(\PageIndex{20}\):

    Simplify: (−12a)2.

    Answer

    144a2

    Example \(\PageIndex{11}\):

    Simplify: (3xy)3.

    Solution

    Raise each factor to the third power. $$3^{\textcolor{red}{3}} x^{\textcolor{red}{3}} y^{\textcolor{red}{3}} \tag{10.2.30}$$
    Simplify. $$27x^{3} y^{3} \tag{10.2.31}$$

    Exercise \(\PageIndex{21}\):

    Simplify: (−4xy)4.

    Answer

    256x4y4

    Exercise \(\PageIndex{22}\):

    Simplify: (6xy)3.

    Answer

    216x3y3

    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.

    Definition: Properties of Exponents

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

    Product Property am • an = am + n
    Power Property (am)n = am • n
    Product to a Power Property (ab)m = ambm

    Example \(\PageIndex{12}\):

    Simplify: (x2)6(x5)4.

    Solution

    Use the Power Property. x12 • x20
    Add the exponents. x32

    Exercise \(\PageIndex{23}\):

    Simplify: (x4)3(x7)4.

    Answer

    x40

    Exercise \(\PageIndex{24}\):

    Simplify: (y9)2(y8)3.

    Answer

    y42

    Example \(\PageIndex{13}\):

    Simplify: (−7x3y4)2.

    Solution

    Take each factor to the second power. (−7)2(x3)2(y4)2
    Use the Power Property. 49x6y8

    Exercise \(\PageIndex{25}\):

    Simplify: (−8x4y7)3.

    Answer

    -512x12y21

    Exercise \(\PageIndex{26}\):

    Simplify: (−3a5b6)4.

    Answer

    81a20b24

    Example \(\PageIndex{14}\):

    Simplify: (6n)2(4n3).

    Solution

    Raise 6n to the second power. 62n2 • 4n3
    Simplify. 36n2 • 4n3
    Use the Commutative Property. 36 • 4 • n2 • n3
    Multiply the constants and add the exponents. 144n5

    Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.

    Exercise \(\PageIndex{27}\):

    Simplify: (7n)2 (2n12).

    Answer

    98n14

    Exercise \(\PageIndex{28}\):

    Simplify: (4m)2(3m3).

    Answer

    48m5

    Example \(\PageIndex{15}\):

    Simplify: (3p2q)4(2pq2)3.

    Solution

    Use the Power of a Product Property. 34(p2)4q4 • 23p3(q2)3
    Use the Power Property. 81p8q4 • 8p3q6
    Use the Commutative Property. 81 • 8 • p8 • p3 • q4 • q6
    Multiply the constants and add the exponents for each variable. 648p11q10

    Exercise \(\PageIndex{29}\):

    Simplify: (u3v2)5(4uv4)3.

    Answer

    64u18v22

    Exercise \(\PageIndex{30}\):

    Simplify: (5x2y3)2(3xy4)3.

    Answer

    675x7y18

    Multiply Monomials

    Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

    Example \(\PageIndex{16}\):

    Multiply: (4x2)(−5x3).

    Solution

    Use the Commutative Property to rearrange the factors. 4 • (−5) • x2 • x3
    Multiply. −20x5

    Exercise \(\PageIndex{31}\):

    Multiply: (7x7)(−8x4).

    Answer

    -56x11

    Exercise \(\PageIndex{32}\):

    Multiply: (−9y4)(−6y5).

    Answer

    54y9

    Example \(\PageIndex{17}\):

    Multiply: \(\left(\dfrac{3}{4} c^{3} d\right)\)(12cd2).

    Solution

    Use the Commutative Property to rearrange the factors. \(\dfrac{3}{4}\) • 12 • c3 • c • d • d2
    Multiply. 9c4d3

    Exercise \(\PageIndex{33}\):

    Multiply: \(\left(\dfrac{4}{5} m^{4} n^{3} d\right)\)(15mn3).

    Answer

    12m5n6

    Exercise \(\PageIndex{34}\):

    Multiply: \(\left(\dfrac{2}{3} p^{5} q d\right)\)(18p6q7).

    Answer

    12p11q8

    ACCESS ADDITIONAL ONLINE RESOURCES

    Exponent Properties

    Exponent Properties 2

    Practice Makes Perfect

    Simplify Expressions with Exponents

    In the following exercises, simplify each expression with exponents.

    1. 45
    2. 103
    3. \(\left(\dfrac{1}{2}\right)^{2}\)
    4. \(\left(\dfrac{3}{5}\right)^{2}\)
    5. (0.2)3
    6. (0.4)3
    7. (−5)4
    8. (−3)5
    9. −54
    10. −35
    11. −104
    12. −26
    13. \(- \left(\dfrac{2}{3}\right)^{3}\)
    14. \(- \left(\dfrac{1}{4}\right)^{4}\)
    15. −0.52
    16. −0.14

    Simplify Expressions Using the Product Property of Exponents

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

    1. x3 • x6
    2. m4 • m2
    3. a • a4
    4. y12 • y
    5. 35 • 39
    6. 510 • 56
    7. z • z2 • z3
    8. a • a3 • a5
    9. xa • x2
    10. yp • y3
    11. ya • yb
    12. xp • xq

    Simplify Expressions Using the Power Property of Exponents

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

    1. (u4)2
    2. (x2)7
    3. (y5)4
    4. (a3)2
    5. (102)6
    6. (28)3
    7. (x15)6
    8. (y12)8
    9. (x2)y
    10. (y3)x
    11. (5x)y
    12. (7a)b

    Simplify Expressions Using the Product to a Power Property

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

    1. (5a)2
    2. (7x)2
    3. (−6m)3
    4. (−9n)3
    5. (4rs)2
    6. (5ab)3
    7. (4xyz)4
    8. (−5abc)3

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify each expression.

    1. (x2)4 • (x3)2
    2. (y4)3 • (y5)2
    3. (a2)6 • (a3)8
    4. (b7)5 • (b2)6
    5. (3x)2(5x)
    6. (2y)3 (6y)
    7. (5a)2(2a)3
    8. (4b)2(3b)3
    9. (2m6)3
    10. (3y2)4
    11. (10x2y)3
    12. (2mn4)5
    13. (−2a3b2)4
    14. (−10u2v4)3
    15. \(\left(\dfrac{2}{3} x^{2} y \right)^{3}\)
    16. \(\left(\dfrac{7}{9} p q^{4} \right)^{2}\)
    17. (8a3)2(2a)4
    18. (5r2)3(3r)2
    19. (10p4)3(5p6)2
    20. (4x3)3(2x5)4
    21. \(\left(\dfrac{1}{2} x^{2} y^{3} \right)^{4}\) (4x5y3)2
    22. \(\left(\dfrac{1}{3} m^{3} n^{2} \right)^{4}\) (9m8n3)2
    23. (3m2n)2(2mn5)4
    24. (2pq4)3(5p6q)2

    Multiply Monomials

    In the following exercises, multiply the following monomials.

    1. (12x2)(−5x4)
    2. (−10y3)(7y2)
    3. (−8u6)(−9u)
    4. (−6c4)(−12c)
    5. \(\left(\dfrac{1}{5} r^{8} \right)\) (20r3)
    6. \(\left(\dfrac{1}{4} a^{5} \right)\) (36a2)
    7. (4a3b)(9a2b6)
    8. (6m4n3)(7mn5)
    9. \(\left(\dfrac{4}{7} x y^{2} \right)\) (14xy3)
    10. \(\left(\dfrac{5}{8} u^{3} v \right)^{3}\) (24u5v)
    11. \(\left(\dfrac{2}{3} x^{2} y \right) \left(\dfrac{3}{4} x y^{2} \right)\)
    12. \(\left(\dfrac{3}{5} m^{3} n^{2} \right) \left(\dfrac{5}{9} m^{2} n^{3} \right)\)

    Everyday Math

    1. Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is 62, on the third round is 63, as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.
    Round Number of people
    1 6
    2 62
    3 63
    … …
    8 ?
    1. Salary Raul’s boss gives him a 5% raise every year on his birthday. This means that each year, Raul’s salary is 1.05 times his last year’s salary. If his original salary was $40,000, his salary after 1 year was $40,000(1.05), after 2 years was $40,000(1.05)2, after 3 years was $40,000(1.05)3, as shown in the table below. What will Raul’s salary be after 10 years? Simplify the expression, to show Raul’s salary in dollars.
    Year Salary
    1 $40,000(1.05)
    2 $40,000(1.05)2
    3 $40,000(1.05)3
    … …
    10 ?

    Writing Exercises

    1. Use the Product Property for Exponents to explain why x • x = x2.
    2. Explain why −53 = (−5)3 but −54 ≠ (−5)4.
    3. Jorge thinks \(\left(\dfrac{1}{2}\right)^{2}\) is 1. What is wrong with his reasoning?
    4. Explain why x3 • x5 is x8, and not x15.

    Self Check

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

    CNX_BMath_Figure_AppB_060.jpg

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

    Contributors and Attributions

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