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10.5 Integer Exponents and Scientific Notation

Negative exponents as reciprocals, simplifying with the full set of exponent rules, and converting to and from scientific notation for very large or tiny numbers.

Prealgebra 2e, 10.5 Integer Exponents and Scientific Notation

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102 exercises

10.5 Exercise 1

Before you get started, take this readiness quiz.

What is the place value of the 66 in the number 64,891?64,891 ?
If you missed this problem, review the reference.

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10.5 Exercise 2

Name the decimal 0.0012.0.0012 .
If you missed this problem, review the reference.

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10.5 Exercise 3

Subtract: 5−(−3).5 - \left(−3\right) .
If you missed this problem, review the reference.

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10.5 Exercise 4

Simplify:

  1. ⓐ 4−24^{−2}
  2. ⓑ 10−310^{−3}

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10.5 Exercise 5

Simplify:

  1. ⓐ 2−32^{−3}
  2. ⓑ 10−210^{−2}

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10.5 Exercise 6

Simplify:

  1. ⓐ 3−23^{−2}
  2. ⓑ 10−410^{−4}

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10.5 Exercise 7

Simplify:

  1. ⓐ (−3)−2\left(−3\right)^{−2}
  2. ⓑ −3−2−3^{−2}

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10.5 Exercise 8

Simplify:

  1. ⓐ (−5)−2\left(−5\right)^{−2}
  2. ⓑ −5−2- 5^{−2}

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10.5 Exercise 9

Simplify:

  1. ⓐ (−2)−2\left(−2\right)^{−2}
  2. ⓑ −2−2−2^{−2}

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10.5 Exercise 10

Simplify:

  1. ⓐ 4·2−14 \cdot 2^{−1}
  2. ⓑ (4·2)−1\left(4 \cdot 2\right)^{−1}

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10.5 Exercise 11

Simplify:

  1. ⓐ 6·3−16 \cdot 3^{−1}
  2. ⓑ (6·3)−1\left(6 \cdot 3\right)^{−1}

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10.5 Exercise 12

Simplify:

  1. ⓐ 8·2−28 \cdot 2^{−2}
  2. ⓑ (8·2)−2\left(8 \cdot 2\right)^{−2}

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10.5 Exercise 13

Simplify: x−6.x^{−6} .

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10.5 Exercise 14

Simplify: y−7.y^{−7} .

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10.5 Exercise 15

Simplify: z−8.z^{−8} .

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10.5 Exercise 16

Simplify:

  1. ⓐ 5y−15 y^{−1}
  2. ⓑ (5y)−1\left(5 y\right)^{−1}
  3. ⓒ (−5y)−1\left(−5 y\right)^{−1}

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10.5 Exercise 17

Simplify:

  1. ⓐ 8p−18 p^{−1}
  2. ⓑ (8p)−1\left(8 p\right)^{−1}
  3. ⓒ (−8p)−1\left(−8 p\right)^{−1}

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10.5 Exercise 18

Simplify:

  1. ⓐ 11q−111 q^{−1}
  2. ⓑ (11q)−1\left(11 q\right)^{−1}
  3. ⓒ (−11q)−1\left(−11 q\right)^{−1}

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10.5 Exercise 19

Simplify:

  1. ⓐ x−4·x6x^{−4} \cdot x^{6}
  2. ⓑ y−6·y4y^{−6} \cdot y^{4}
  3. ⓒ z−5·z−3z^{−5} \cdot z^{−3}

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10.5 Exercise 20

Simplify:

  1. ⓐ x−3·x7x^{−3} \cdot x^{7}
  2. ⓑ y−7·y2y^{−7} \cdot y^{2}
  3. ⓒ z−4·z−5z^{−4} \cdot z^{−5}

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10.5 Exercise 21

Simplify:

  1. ⓐ a−1·a6a^{−1} \cdot a^{6}
  2. ⓑ b−8·b4b^{−8} \cdot b^{4}
  3. ⓒ c−8·c−7c^{−8} \cdot c^{−7}

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10.5 Exercise 22

Simplify: (m4n−3)(m−5n−2).\left(m^{4} n^{−3}\right) \left(m^{−5} n^{−2}\right) .

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10.5 Exercise 23

Simplify: (p6q−2)(p−9q−1).\left(p^{6} q^{−2}\right) \left(p^{−9} q^{−1}\right) .

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10.5 Exercise 24

Simplify: (r5s−3)(r−7s−5).\left(r^{5} s^{−3}\right) \left(r^{−7} s^{−5}\right) .

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10.5 Exercise 25

Simplify: (2x−6y8)(−5x5y−3).\left(2 x^{−6} y^{8}\right) \left(−5 x^{5} y^{−3}\right) .

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10.5 Exercise 26

Simplify: (3u−5v7)(−4u4v−2).\left(3 u^{−5} v^{7}\right) \left(−4 u^{4} v^{−2}\right) .

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10.5 Exercise 27

Simplify: (−6c−6d4)(−5c−2d−1).\left(−6 c^{−6} d^{4}\right) \left(−5 c^{−2} d^{−1}\right) .

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10.5 Exercise 28

Simplify: (k3)−2.\left(k^{3}\right)^{−2} .

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10.5 Exercise 29

Simplify: (x4)−1.\left(x^{4}\right)^{−1} .

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10.5 Exercise 30

Simplify: (y2)−2.\left(y^{2}\right)^{−2} .

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10.5 Exercise 31

Simplify: (5x−3)2.\left(5 x^{−3}\right)^{2} .

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10.5 Exercise 32

Simplify: (8a−4)2.\left(8 a^{−4}\right)^{2} .

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10.5 Exercise 33

Simplify: (2c−4)3.\left(2 c^{−4}\right)^{3} .

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10.5 Exercise 35

Simplify: x8x−3.\frac{x^{8}}{x^{−3}} .

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10.5 Exercise 36

Simplify: y7y−6.\frac{y^{7}}{y^{−6}} .

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10.5 Exercise 38

Write in scientific notation: 96,000.96,000 .

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10.5 Exercise 39

Write in scientific notation: 48,300.48,300 .

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10.5 Exercise 41

Write in scientific notation: 0.0078.0.0078 .

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10.5 Exercise 42

Write in scientific notation: 0.0129.0.0129 .

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10.5 Exercise 44

Convert to decimal form: 1.3×103.1.3 \times 10^{3} .

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10.5 Exercise 45

Convert to decimal form: 9.25×104.9.25 \times 10^{4} .

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10.5 Exercise 47

Convert to decimal form: 1.2×10−4.1.2 \times 10^{−4} .

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10.5 Exercise 48

Convert to decimal form: 7.5×10−2.7.5 \times 10^{−2} .

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10.5 Exercise 49

Multiply. Write answers in decimal form: (4×105)(2×10−7).\left(4 \times 10^{5}\right) \left(2 \times 10^{−7}\right) .

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10.5 Exercise 50

Multiply. Write answers in decimal form: (3×106)(2×10−8).\left(3 \times 10^{6}\right) \left(2 \times 10^{−8}\right) .

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10.5 Exercise 51

Multiply. Write answers in decimal form: (3×10−2)(3×10−1).\left(3 \times 10^{−2}\right) \left(3 \times 10^{−1}\right) .

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10.5 Exercise 52

Divide. Write answers in decimal form: 9×1033×10−2.\frac{9 \times 10^{3}}{3 \times 10^{−2}} .

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10.5 Exercise 53

Divide. Write answers in decimal form: 8×1042×10−1.\frac{8 \times 10^{4}}{2 \times 10^{−1}} .

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10.5 Exercise 54

Divide. Write answers in decimal form: 8×1024×10−2.\frac{8 \times 10^{2}}{4 \times 10^{−2}} .

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10.5 Exercise 56

Simplify.

8−28^{−2}

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10.5 Exercise 58

Simplify.

2−52^{−5}

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10.5 Exercise 60

Simplify.

10−110^{−1}

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10.5 Exercise 62

Simplify.

3−2+3−13^{−2} + 3^{−1}

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10.5 Exercise 64

Simplify.

10−1+2−110^{−1} + 2^{−1}

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10.5 Exercise 66

Simplify.

20−2−1+2−22^{0} - 2^{−1} + 2^{−2}

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10.5 Exercise 68

Simplify.

  1. ⓐ (−8)−2\left(−8\right)^{−2}
  2. ⓑ −8−2- 8^{−2}

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10.5 Exercise 70

Simplify.

  1. ⓐ (−4)−6\left(−4\right)^{−6}
  2. ⓑ −4−6- 4^{−6}

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10.5 Exercise 72

Simplify.

  1. ⓐ 10·3−110 \cdot 3^{−1}
  2. ⓑ (10·3)−1\left(10 \cdot 3\right)^{−1}

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10.5 Exercise 74

Simplify.

  1. ⓐ 3·5−23 \cdot 5^{−2}
  2. ⓑ (3·5)−2\left(3 \cdot 5\right)^{−2}

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10.5 Exercise 76

Simplify.

p−3p^{−3}

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10.5 Exercise 78

Simplify.

m−5m^{−5}

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10.5 Exercise 80

Simplify.

  1. ⓐ 3q−13 q^{−1}
  2. ⓑ (3q)−1\left(3 q\right)^{−1}
  3. ⓒ (−3q)−1\left(−3 q\right)^{−1}

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10.5 Exercise 82

Simplify.

  1. ⓐ 10k−110 k^{−1}
  2. ⓑ (10k)−1\left(10 k\right)^{−1}
  3. ⓒ (−10k)−1\left(−10 k\right)^{−1}

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10.5 Exercise 84

Simplify.

r−2·r5r^{−2} \cdot r^{5}

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10.5 Exercise 86

Simplify.

q−8·q3q^{−8} \cdot q^{3}

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10.5 Exercise 88

Simplify.

z−6·z−2z^{−6} \cdot z^{−2}

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10.5 Exercise 90

Simplify.

m·m−2m \cdot m^{−2}

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10.5 Exercise 92

Simplify.

x4·x−2·x−3x^{4} \cdot x^{−2} \cdot x^{−3}

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10.5 Exercise 94

Simplify.

u2v−2u^{2} v^{−2}

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10.5 Exercise 96

Simplify.

(a3b−3)(a−5b−1)\left(a^{3} b^{−3}\right) \left(a^{−5} b^{−1}\right)

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10.5 Exercise 98

Simplify.

(pq−4)(p−6q−3)\left(p q^{−4}\right) \left(p^{−6} q^{−3}\right)

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10.5 Exercise 100

Simplify.

(−3p−5q8)(7p2q−3)\left(−3 p^{−5} q^{8}\right) \left(7 p^{2} q^{−3}\right)

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10.5 Exercise 102

Simplify.

(−8a−5b−4)(−4a2b3)\left(−8 a^{−5} b^{−4}\right) \left(−4 a^{2} b^{3}\right)

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10.5 Exercise 104

Simplify.

(q10)−10\left(q^{10}\right)^{−10}

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10.5 Exercise 106

Simplify.

(x4)−1\left(x^{4}\right)^{−1}

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10.5 Exercise 108

Simplify.

(p−3)2\left(p^{−3}\right)^{2}

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10.5 Exercise 110

Simplify.

(m−2)−3\left(m^{−2}\right)^{−3}

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10.5 Exercise 112

Simplify.

(3q−5)2\left(3 q^{−5}\right)^{2}

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10.5 Exercise 114

Simplify.

(2n−3)−6\left(2 n^{−3}\right)^{−6}

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10.5 Exercise 116

Simplify.

b5b−3\frac{b^{5}}{b^{−3}}

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10.5 Exercise 118

Simplify.

m5m−2\frac{m^{5}}{m^{−2}}

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10.5 Exercise 120

Simplify.

r6r9\frac{r^{6}}{r^{9}}

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10.5 Exercise 122

Simplify.

p−3p−6\frac{p^{−3}}{p^{−6}}

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10.5 Exercise 124

Write each number in scientific notation.

280,000

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10.5 Exercise 126

Write each number in scientific notation.

1,290,000

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10.5 Exercise 128

Write each number in scientific notation.

0.041

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10.5 Exercise 130

Write each number in scientific notation.

0.0000103

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10.5 Exercise 132

Write each number in scientific notation.

The population of the world on July 4, 2010 was more than 6,850,000,000.6,850,000,000 .

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10.5 Exercise 134

Write each number in scientific notation.

The probability of winning the 20102010 Megamillions lottery is about 0.0000000057.0.0000000057 .

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10.5 Exercise 136

Convert each number to decimal form.

8.3×1028.3 \times 10^{2}

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10.5 Exercise 138

Convert each number to decimal form.

1.6×10101.6 \times 10^{10}

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10.5 Exercise 140

Convert each number to decimal form.

2.8×10−22.8 \times 10^{−2}

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10.5 Exercise 142

Convert each number to decimal form.

6.15×10−86.15 \times 10^{−8}

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10.5 Exercise 144

Convert each number to decimal form.

At the start of 2012, the US federal budget had a deficit of more than $1.5×1013.\$\text{1}.\text{5} \times 10^{13} .

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10.5 Exercise 146

Convert each number to decimal form.

The width of a proton is 1×10−51 \times 10^{−5} of the width of an atom.

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10.5 Exercise 148

Multiply or divide and write your answer in decimal form.

(3×102)(1×10−5)\left(3 \times 10^{2}\right) \left(1 \times 10^{−5}\right)

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10.5 Exercise 150

Multiply or divide and write your answer in decimal form.

(2.1×10−4)(3.5×10−2)\left(2.1 \times 10^{−4}\right) \left(3.5 \times 10^{−2}\right)

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10.5 Exercise 152

Multiply or divide and write your answer in decimal form.

8×1064×10−1\frac{8 \times 10^{6}}{4 \times 10^{−1}}

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10.5 Exercise 154

Multiply or divide and write your answer in decimal form.

5×10−31×10−10\frac{5 \times 10^{−3}}{1 \times 10^{−10}}

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10.5 Exercise 156

Length of a year The difference between the calendar year and the astronomical year is 0.0001250.000125 day.

  1. ⓐ Write this number in scientific notation.
  2. ⓑ How many years does it take for the difference to become 1 day?
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10.5 Exercise 158

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with 66 of her 5050 favorite photographs, she multiplied 50·49·48·47·46·45.50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 . Her calculator gave the answer 1.1441304×1010.1.1441304 \times 10^{10} . Write the number in decimal notation.

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10.5 Exercise 160

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

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