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Mathematics LibreTexts

6.2E: Exercises

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    Practice Makes Perfect

    Simplify Expressions with Exponents

    In the following exercises, simplify each expression with exponents.

    Exercise 1

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

    Exercise 2

    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 3

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

    Exercise 4

    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 5

    1. \((−6)^4\)
    2. \(−6^4\)

    Exercise 6

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

    Exercise 7

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

    Exercise 8

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

    Exercise 9

    1. \(−0.5^2\)
    2. \((−0.5)^2\)

    Exercise 10

    1. \(−0.1^4\)
    2. \((−0.1)^4\)
    Answer
    1. −0.0001
    2. 0.0001

    Simplify Expressions Using the Product Property for Exponents

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

    Exercise 11

    \(d^3·d^6\)

    Exercise 12

    \(x^4·x^2\)

    Answer

    \(x^6\)

    Exercise 13

    \(n^{19}·n^{12}\)

    Exercise 14

    \(q^{27}·q^{15}\)

    Answer

    \(q^{42}\)

    Exercise 15

    1. \(4^5·4^9\)
    2. \(8^9·8\)

    Exercise 16

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

    Exercise 17

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

    Exercise 18

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

    Exercise 19

    \(w·w^2·w^3\)

    Exercise 20

    \(y·y^3·y^5\)

    Answer

    \(y^9\)

    Exercise 21

    \(a^4·a^3·a^9\)

    Exercise 22

    \(c^5·c^{11}·c^2\)

    Answer

    \(c^{18}\)

    Exercise 23

    \(m^x·m^3\)

    Exercise 24

    \(n^y·n^2\)

    Answer

    \(n^{y+2}\)

    Exercise 25

    \(y^a·y^b\)

    Exercise 26

    \(x^p·x^q\)

    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 27

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

    Exercise 28

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

    Exercise 29

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

    Exercise 30

    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 31

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

    Exercise 32

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

    Exercise 33

    1. \((−4m)^3\)
    2. \((5ab)^3\)

    Exercise 34

    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 35

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

    Exercise 36

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

    Exercise 37

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

    Exercise 38

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

    Exercise 39

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

    Exercise 40

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

    Exercise 41

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

    Exercise 42

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

    Exercise 43

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

    Exercise 44

    1. \((2r^2)^{3}(4r)^2\)
    2. \((3x^3)^{3}(x^5)^4\)
    Answer
    1. \(128r^{8}\)
    2. \(27x^{29}\)

    Exercise 45

    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 46

    \((6y^7)(−3y^4)\)

    Answer

    \(−18y^{11}\)

    Exercise 47

    \((−10x^5)(−3x^3)\)

    Exercise 48

    \((−8u^6)(−9u)\)

    Answer

    \(72u^{7}\)

    Exercise 49

    \((−6c^4)(−12c)\)

    Exercise 50

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

    Answer

    \(4f^{11}\)

    Exercise 51

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

    Exercise 52

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

    Answer

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

    Exercise 53

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

    Exercise 54

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

    Answer

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

    Exercise 55

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

    Exercise 56

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

    Answer

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

    Exercise 57

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

    ​​​​​​Mixed Practice

    In the following exercises, simplify each expression.

    Exercise 58

    \((x^2)^4·(x^3)^2\)

    Answer

    \(x^{14}\)

    Exercise 59

    \((y^4)^3·(y^5)^2\)

    Exercise 60

    \((a^2)^6·(a^3)^8\)

    Answer

    \(a^{36}\)

    Exercise 61

    \((b^7)^5·(b^2)^6\)

    Exercise 62

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

    Answer

    \(8m^{18}\)

    Exercise 63

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

    Exercise 64

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

    Answer

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

    Exercise 65

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

    Exercise 66

    \((−2a^{3}b^2)^4\)

    Answer

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

    Exercise 67

    \((−10u^{2}v^4)^3\)

    Exercise 68

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

    Answer

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

    Exercise 69

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

    Exercise 70

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

    Answer

    \(1024a^{10}\)

    Exercise 71

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

    Exercise 72

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

    Answer

    \(25000p^{24}\)

    Exercise 73

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

    Exercise 74

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

    Answer

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

    Exercise 75

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

    Exercise 76

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

    Answer

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

    Exercise 77

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

    ​​​​​​Everyday Math

    Exercise 78

    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.

    Round Number of People
    1 10
    2 \(10^2\)
    3 \(10^3\)
    ... ...
    6 ?
    Answer

    1,000,000

    Exercise 79

    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.

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

    Exercise 80

    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.

    Week Cost
    1 $1,000(0.70)
    2 $\(1,000(0.70)^2\)
    3
    4 ...
    5 ?
    Answer

    $168.07

    Exercise 81

    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.

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

    Writing Exercises

    Exercise 82

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

    Answer

    Answers will vary.

    Exercise 83

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

    Exercise 84

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

    Answer

    Answers will vary.

    Exercise 85

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

    Self Check

    a. 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.

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