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6.2 Use Multiplication Properties of Exponents

Product, Power, and Product-to-a-Power rules for exponents, chained together on messy expressions and applied to multiplying monomials.

Elementary Algebra 2e, 6.2 Use Multiplication Properties of Exponents

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74 exercises

6.2 Exercise 1

Before you get started, take this readiness quiz.

Simplify: 34·34.\frac{3}{4} \cdot \frac{3}{4} .
If you missed this problem, review the reference.

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6.2 Exercise 2

Simplify: (−2)(−2)(−2).\left(−2\right) \left(−2\right) \left(−2\right) .
If you missed this problem, review the reference.

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6.2 Exercise 3

Simplify: ⓐ 434^{3} ⓑ 717^{1} ⓒ (56)2\left(\frac{5}{6}\right)^{2} ⓓ (0.63)2.\left(0.63\right)^{2} .

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6.2 Exercise 4

Simplify: ⓐ 636^{3} ⓑ 15115^{1} ⓒ (37)2\left(\frac{3}{7}\right)^{2} ⓓ (0.43)2.\left(0.43\right)^{2} .

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6.2 Exercise 5

Simplify: ⓐ 252^{5} ⓑ 21121^{1} ⓒ (25)3\left(\frac{2}{5}\right)^{3} ⓓ (0.218)2.\left(0.218\right)^{2} .

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6.2 Exercise 6

Simplify: ⓐ (−5)4\left(−5\right)^{4} ⓑ −54.− 5^{4} .

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6.2 Exercise 7

Simplify: ⓐ (−3)4\left(−3\right)^{4} ⓑ −34.− 3^{4} .

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6.2 Exercise 8

Simplify: ⓐ (−13)2\left(−13\right)^{2} ⓑ −132.− 13^{2} .

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6.2 Exercise 10

Simplify: b9·b8.b^{9} \cdot b^{8} .

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6.2 Exercise 11

Simplify: x12·x4.x^{12} \cdot x^{4} .

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6.2 Exercise 13

Simplify: ⓐ 5·555 \cdot 5^{5} ⓑ 49·49.4^{9} \cdot 4^{9} .

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6.2 Exercise 14

Simplify: ⓐ 76·787^{6} \cdot 7^{8} ⓑ 10·1010.10 \cdot 10^{10} .

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6.2 Exercise 16

Simplify: ⓐ p5·pp^{5} \cdot p ⓑ y14·y29.y^{14} \cdot y^{29} .

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6.2 Exercise 17

Simplify: ⓐ z·z7z \cdot z^{7} ⓑ b15·b34.b^{15} \cdot b^{34} .

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6.2 Exercise 19

Simplify: x6·x4·x8.x^{6} \cdot x^{4} \cdot x^{8} .

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6.2 Exercise 20

Simplify: b5·b9·b5.b^{5} \cdot b^{9} \cdot b^{5} .

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6.2 Exercise 22

Simplify: ⓐ (b7)5\left(b^{7}\right)^{5} ⓑ (54)3.\left(5^{4}\right)^{3} .

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6.2 Exercise 23

Simplify: ⓐ (z6)9\left(z^{6}\right)^{9} ⓑ (37)7.\left(3^{7}\right)^{7} .

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6.2 Exercise 25

Simplify: ⓐ (−12y)2\left(−12 y\right)^{2} ⓑ (2wx)5.\left(2 w x\right)^{5} .

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6.2 Exercise 26

Simplify: ⓐ (5wx)3\left(5 w x\right)^{3} ⓑ (−3y)3.\left(−3 y\right)^{3} .

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6.2 Exercise 27

Simplify: ⓐ (y3)6(y5)4\left(y^{3}\right)^{6} \left(y^{5}\right)^{4} ⓑ (−6x4y5)2.\left(−6 x^{4} y^{5}\right)^{2} .

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6.2 Exercise 28

Simplify: ⓐ (a4)5(a7)4\left(a^{4}\right)^{5} \left(a^{7}\right)^{4} ⓑ (−2c4d2)3.\left(−2 c^{4} d^{2}\right)^{3} .

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6.2 Exercise 29

Simplify: ⓐ (−3x6y7)4\left(−3 x^{6} y^{7}\right)^{4} ⓑ (q4)5(q3)3.\left(q^{4}\right)^{5} \left(q^{3}\right)^{3} .

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6.2 Exercise 30

Simplify: ⓐ (5m)2(3m3)\left(5 m\right)^{2} \left(3 m^{3}\right) ⓑ (3x2y)4(2xy2)3.\left(3 x^{2} y\right)^{4} \left(2 x y^{2}\right)^{3} .

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6.2 Exercise 31

Simplify: ⓐ (5n)2(3n10)\left(5 n\right)^{2} \left(3 n^{10}\right) ⓑ (c4d2)5(3cd5)4.\left(c^{4} d^{2}\right)^{5} \left(3 c d^{5}\right)^{4} .

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6.2 Exercise 32

Simplify: ⓐ (a3b2)6(4ab3)4\left(a^{3} b^{2}\right)^{6} \left(4 a b^{3}\right)^{4} ⓑ (2x)3(5x7).\left(2 x\right)^{3} \left(5 x^{7}\right) .

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6.2 Exercise 33

Multiply: (3x2)(−4x3).\left(3 x^{2}\right) \left(−4 x^{3}\right) .

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6.2 Exercise 34

Multiply: (5y7)(−7y4).\left(5 y^{7}\right) \left(−7 y^{4}\right) .

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6.2 Exercise 35

Multiply: (−6b4)(−9b5).\left(−6 b^{4}\right) \left(−9 b^{5}\right) .

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6.2 Exercise 36

Multiply: (56x3y)(12xy2).\left(\frac{5}{6} x^{3} y\right) \left(12 x y^{2}\right) .

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6.2 Exercise 37

Multiply: (25a4b3)(15ab3).\left(\frac{2}{5} a^{4} b^{3}\right) \left(15 a b^{3}\right) .

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6.2 Exercise 38

Multiply: (23r5s)(12r6s7).\left(\frac{2}{3} r^{5} s\right) \left(12 r^{6} s^{7}\right) .

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6.2 Exercise 40

Simplify each expression with exponents.

ⓐ 10410^{4} ⓑ 17117^{1} ⓒ (29)2\left(\frac{2}{9}\right)^{2} ⓓ (0.5)3\left(0.5\right)^{3}

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6.2 Exercise 42

Simplify each expression with exponents.

ⓐ 838^{3} ⓑ 818^{1} ⓒ (34)3\left(\frac{3}{4}\right)^{3} ⓓ (0.4)3\left(0.4\right)^{3}

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6.2 Exercise 44

Simplify each expression with exponents.

ⓐ (−2)6\left(−2\right)^{6} ⓑ −26− 2^{6}

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6.2 Exercise 46

Simplify each expression with exponents.

ⓐ −(23)2− \left(\frac{2}{3}\right)^{2} ⓑ (−23)2\left(- \frac{2}{3}\right)^{2}

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6.2 Exercise 48

Simplify each expression with exponents.

ⓐ −0.14− 0.1^{4} ⓑ (−0.1)4\left(−0.1\right)^{4}

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6.2 Exercise 50

Simplify each expression using the Product Property for Exponents.

x4·x2x^{4} \cdot x^{2}

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6.2 Exercise 52

Simplify each expression using the Product Property for Exponents.

q27·q15q^{27} \cdot q^{15}

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6.2 Exercise 54

Simplify each expression using the Product Property for Exponents.

ⓐ 310·363^{10} \cdot 3^{6} ⓑ 5·545 \cdot 5^{4}

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6.2 Exercise 56

Simplify each expression using the Product Property for Exponents.

ⓐ w5·ww^{5} \cdot w^{} ⓑ u41·u53u^{41} \cdot u^{53}

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6.2 Exercise 58

Simplify each expression using the Product Property for Exponents.

y·y3·y5y \cdot y^{3} \cdot y^{5}

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6.2 Exercise 60

Simplify each expression using the Product Property for Exponents.

c5·c11·c2c^{5} \cdot c^{11} \cdot c^{2}

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6.2 Exercise 62

Simplify each expression using the Product Property for Exponents.

ny·n2n^{y} \cdot n^{2}

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6.2 Exercise 64

Simplify each expression using the Product Property for Exponents.

xp·xqx^{p} \cdot x^{q}

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6.2 Exercise 66

Simplify each expression using the Power Property for Exponents.

ⓐ (b2)7\left(b^{2}\right)^{7} ⓑ (38)2\left(3^{8}\right)^{2}

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6.2 Exercise 68

Simplify each expression using the Power Property for Exponents.

ⓐ (x2)y\left(x^{2}\right)^{y} ⓑ (7a)b\left(7^{a}\right)^{b}

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6.2 Exercise 70

Simplify each expression using the Product to a Power Property.

ⓐ (5x)2\left(5 x\right)^{2} ⓑ (4ab)2\left(4 a b\right)^{2}

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6.2 Exercise 72

Simplify each expression using the Product to a Power Property.

ⓐ (−7n)3\left(−7 n\right)^{3} ⓑ (3xyz)4\left(3 x y z\right)^{4}

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6.2 Exercise 74

Simplify each expression.

ⓐ (w4)3·(w5)2\left(w^{4}\right)^{3} \cdot \left(w^{5}\right)^{2} ⓑ (2xy4)5\left(2 x y^{4}\right)^{5}

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6.2 Exercise 76

Simplify each expression.

ⓐ (−10q2p4)3\left(−10 q^{2} p^{4}\right)^{3} ⓑ (n3)10·(n5)2\left(n^{3}\right)^{10} \cdot \left(n^{5}\right)^{2}

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6.2 Exercise 78

Simplify each expression.

ⓐ (2y)3(6y)\left(2 y\right)^{3} \left(6 y\right) ⓑ (10k4)3(5k6)2\left(10 k^{4}\right)^{3} \left(5 k^{6}\right)^{2}

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6.2 Exercise 80

Simplify each expression.

ⓐ (4b)2(3b)3\left(4 b\right)^{2} \left(3 b\right)^{3} ⓑ (12j2)5(25j3)2\left(\frac{1}{2} j^{2}\right)^{5} \left(\frac{2}{5} j^{3}\right)^{2}

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6.2 Exercise 82

Simplify each expression.

ⓐ (2r2)3(4r)2\left(2 r^{2}\right)^{3} \left(4 r\right)^{2} ⓑ (3x3)3(x5)4\left(3 x^{3}\right)^{3} \left(x^{5}\right)^{4}

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6.2 Exercise 84

Multiply the monomials.

(6y7)(−3y4)\left(6 y^{7}\right) \left(−3 y^{4}\right)

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6.2 Exercise 86

Multiply the monomials.

(−8u6)(−9u)\left(−8 u^{6}\right) \left(−9 u\right)

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6.2 Exercise 88

Multiply the monomials.

(15f8)(20f3)\left(\frac{1}{5} f^{8}\right) \left(20 f^{3}\right)

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6.2 Exercise 90

Multiply the monomials.

(4a3b)(9a2b6)\left(4 a^{3} b\right) \left(9 a^{2} b^{6}\right)

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6.2 Exercise 92

Multiply the monomials.

(47rs2)(14rs3)\left(\frac{4}{7} r s^{2}\right) \left(14 r s^{3}\right)

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6.2 Exercise 94

Multiply the monomials.

(23x2y)(34xy2)\left(\frac{2}{3} x^{2} y\right) \left(\frac{3}{4} x y^{2}\right)

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6.2 Exercise 96

Simplify each expression.

(x2)4·(x3)2\left(x^{2}\right)^{4} \cdot \left(x^{3}\right)^{2}

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6.2 Exercise 98

Simplify each expression.

(a2)6·(a3)8\left(a^{2}\right)^{6} \cdot \left(a^{3}\right)^{8}

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6.2 Exercise 100

Simplify each expression.

(2m6)3\left(2 m^{6}\right)^{3}

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6.2 Exercise 102

Simplify each expression.

(10x2y)3\left(10 x^{2} y\right)^{3}

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6.2 Exercise 104

Simplify each expression.

(−2a3b2)4\left(−2 a^{3} b^{2}\right)^{4}

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6.2 Exercise 106

Simplify each expression.

(23x2y)3\left(\frac{2}{3} x^{2} y\right)^{3}

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6.2 Exercise 108

Simplify each expression.

(8a3)2(2a)4\left(8 a^{3}\right)^{2} \left(2 a\right)^{4}

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6.2 Exercise 110

Simplify each expression.

(10p4)3(5p6)2\left(10 p^{4}\right)^{3} \left(5 p^{6}\right)^{2}

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6.2 Exercise 112

Simplify each expression.

(12x2y3)4(4x5y3)2\left(\frac{1}{2} x^{2} y^{3}\right)^{4} \left(4 x^{5} y^{3}\right)^{2}

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6.2 Exercise 114

Simplify each expression.

(3m2n)2(2mn5)4\left(3 m^{2} n\right)^{2} \left(2 m n^{5}\right)^{4}

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6.2 Exercise 116

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 10210^{2}, on the third round is 10310^{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…

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6.2 Exercise 118

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)\$ 1,000 \left(0.70\right) and the cost of the item during the second week would be $1,000(0.70)2\$ 1,000 \left(0.70\right)^{2}. Complete the table shown belo…

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6.2 Exercise 120

Use the Product Property for Exponents to explain why x·x=x2x \cdot x = x^{2}.

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6.2 Exercise 122

Jorge thinks (12)2\left(\frac{1}{2}\right)^{2} is 1. What is wrong with his reasoning?

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.