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6.7E: Exercises

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Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Exercise 1
  1. 42
  2. 103
Exercise 2
  1. 34
  2. 102
Answer
  1. 181
  2. 1100
Exercise 3
  1. 53
  2. 105
Exercise 4
  1. 28
  2. 102
Answer
  1. 1256
  2. 1100
Exercise 5
  1. 1c5
  2. 132
Exercise 6
  1. 1c5
  2. 152
Answer
  1. c5
  2. 25
Exercise 7
  1. 1q10
  2. 1103
Exercise 8
  1. 1t9
  2. 1104
Answer
  1. t9
  2. 10000
Exercise 9
  1. (58)2
  2. (3mn)2
Exercise 10
  1. (310)2
  2. (2cd)3
Answer
  1. 1009
  2. d38c3
Exercise 11
  1. (49)3
  2. (u22v)5
Exercise 12
  1. (72)3
  2. (3xy2)3
Answer
  1. 8343
  2. x3y627
Exercise 13
  1. (5)2
  2. 52
  3. (15)2
  4. (15)2
Exercise 14
  1. (7)2
  2. 72
  3. (17)2
  4. (17)2
Answer
  1. 149
  2. 149
  3. 49
  4. −49
Exercise 15
  1. 33
  2. (13)3
  3. (13)3
  4. (3)3
Exercise 16
  1. 53
  2. (15)3
  3. (15)3
  4. (5)3
Answer
  1. 1125
  2. −125
  3. −125
  4. 1125
Exercise 17
  1. 3·51
  2. (3·5)1
Exercise 18
  1. 2·51
  2. (2·5)1
Answer
  1. 25
  2. 110
Exercise 19
  1. 4·52
  2. (4·5)2
Exercise 20
  1. 3·42
  2. (3·4)2
Answer
  1. 316
  2. 1144
Exercise 21
  1. m4
  2. (x3)4
Exercise 22
  1. b5
  2. (k2)5
Answer
  1. 1b5
  2. 1k10
Exercise 23
  1. p10
  2. (q6)8
Exercise 24
  1. s8
  2. (a9)10
Answer
  1. 1s8
  2. 1a90
Exercise 25
  1. 7n1
  2. (7n)1
  3. (7n)1
Exercise 26
  1. 6r1
  2. (6r)1
  3. (6r)1
Answer
  1. 6r
  2. 16r
  3. 16r
Exercise 27
  1. (3p)2
  2. 3p2
  3. 3p2
Exercise 28
  1. (2q)4
  2. 2q4
  3. 2q4
Answer
  1. 116q4
  2. 2q4
  3. 2q4

​​​​​Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Exercise 29
  1. b4b8
  2. r2r5
  3. x7x3
Exercise 30
  1. s3·s7
  2. q8·q3
  3. y2·y5
Answer
  1. 1s4
  2. 1q5
  3. 1y7
Exercise 31
  1. a3·a3
  2. a·a3
  3. a·a3
Exercise 32
  1. y5·y5
  2. y·y5
  3. y·y5
Answer
  1. 1
  2. y6
  3. 1y4
Exercise 33

p5·p2·p4

Exercise 34

x4·x2·x3

Answer

1x

Exercise 35

(w4x5)(w2x4)

Exercise 36

(m3n3)(m5n1)

Answer

1m2n4

Exercise 37

(uv2)(u5v3)

Exercise 38

(pq4)(p6q3)

Answer

1p5q7

Exercise 39

(6c3d9)(2c4d5)

Exercise 40

(2j5k8)(7j2k3)

Answer

14k5j3

Exercise 41

(4r2s8)(9r4s3)

Exercise 42

(5m4n6)(8m5n3)

Answer

40n3m

Exercise 43

(5x2)2

Exercise 44

(4y3)3

Answer

164y9

Exercise 45

(3z3)2

Exercise 46

(2p5)2

Answer

4p10

Exercise 47

t9t3

Exercise 48

n5n2

Answer

n7

Exercise 49

x7x3

Exercise 50

y5y10

Answer

y5

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

Exercise 51

57,000

Exercise 52

340,000

Answer

3.4×105

Exercise 53

8,750,000

Exercise 54

1,290,000

Answer

1.29×106

Exercise 55

0.026

Exercise 56

0.041

Answer

4.1×102

Exercise 57

0.00000871

Exercise 58

0.00000103

Answer

1.03×106

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

Exercise 59

5.2×102

Exercise 60

8.3×102

Answer

830

Exercise 61

7.5×106

Exercise 62

1.6×1010

Answer

16,000,000,000

Exercise 63

2.5×102

Exercise 64

3.8×102

Answer

0.038

Exercise 65

4.13×105

Exercise 66

1.93×105

Answer

0.0000193

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

Exercise 67

(3×105)(3×109)

Exercise 68

(2×102)(1×104)

Answer

0.02

Exercise 69

(7.1×102)(2.4×104)

Exercise 70

(3.5×104)(1.6×102)

Answer

5.6×106

In the following exercises, divide. Write your answer in decimal form.

Exercise 71

7×1031×107

Exercise 72

5×1021×1010

Answer

500,000,000

Exercise 73

6×1043×102

Exercise 74

8×1064×101

Answer

20,000,000

Everyday Math

Exercise 75

The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.

Exercise 76

The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation

Answer

6.85×109

Exercise 77

The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.

Exercise 78

The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.

Answer

5.7×109

Exercise 79

In 2010, the number of Facebook users each day who changed their status to 'engaged" was 2×104. Convert this number
to decimal form.

Exercise 80

At the start of 2012, the US federal budget had a deficit of more than $1.5×1013. Convert this number to decimal form.

Answer

15,000,000,000,000

Exercise 81

The concentration of carbon dioxide in the atmosphere is 3.9×104. Convert this number to decimal form.

Exercise 82

The width of a proton is 1×105 of the width of an atom. Convert this number to decimal form.

Answer

0.00001

Exercise 83

Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017.

  1. Write 4 trillion in decimal notation.
  2. Write 4 trillion in scientific notation.
Exercise 84

Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

Answer

1.545×108

Exercise 85

Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles.

  1. Write the number of miles in one light year in scientific notation.
  2. Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation.
Exercise 86

Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.

  1. Write the debt in scientific notation.
  2. Write the population in scientific notation.
  3. Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.
Answer
  1. 1.86×1013
  2. 3×108
  3. 6.2×104

Writing Exercises

Exercise 87
  1. Explain the meaning of the exponent in the expression 23.
  2. Explain the meaning of the exponent in the expression 23.
Exercise 88

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answer

answers will vary

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 six 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 “use the definition of a negative exponent,” “simplify expressions with integer exponents,” “convert from decimal notation to scientific notation,” “convert scientific notation to decimal form,” and “multiply and divide using scientific notation.” The rest of the cells are blank.

b. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

 

More Practice with Exponents in Quotients

Quotients

Exercise 6.7E.89

  Simplify. (Assume all variables represent nonzero numbers.)

1.    102104105

2.    757972

3.    a8a6a5

4.    b4b10b8

5.    a8a3a6

6.    b10b4b2

7.    25x3y25x1y3

8.    9x1y3z53x2y2z1

9.    40x5y3z4x2y2z

10.    8x2y5z316x2yz

11.    24a8b3(a5b)108a5b3(a5b)2

12.    175m9n5(m+n)725m8n(m+n)3

Answers to odd exercises.

1. 10

3. a9

5. a11

7. 5y5x2

9. 10x3y

11. 3a3(a5b)8

Powers of Quotients

Exercise 6.7E.90 

  Simplify. (Assume all variables represent nonzero numbers.)  

1.    (3ab22c3)3

2.    (10a3b3c2)2

3.    (2xy4z3)4

4.    (7x9yz4)3

5.    (12x3y2z2x7yz8)3

6.    (150xy8z290x7y2z)2

7.    (2x3zy2)5

8.    (5x5z22y3)3

9.    (xy2z3)3

10.    (2x2y3z)4

11.    (9a3b4c23a3b5c7)4

12.    (15a7b5c83a6b2c3)3

Answers to odd exercises:

1. 27a3b68c9

3. 16x4y16z12

5. 216y3x12z21

7. x15y1032z5

9. x3y6z9

11. a24b481c20

 


This page titled 6.7E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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