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Key concepts

  • Summary of Exponent Properties
    • If a , b are real numbers and m , n are integers, then
      Product Property a m · a n = a m + n Power Property ( a m ) n = a m · n Product to a Power Property ( a b ) m = a m b m Quotient Property a m a n = a m n , a 0 Zero Exponent Property a 0 = 1 , a 0 Quotient to a Power Property ( a b ) m = a m b m , b 0 Definition of Negative Exponent a n = 1 a n
  • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, n , that the decimal point was moved.
    3. Write the number as a product with a power of 10.
      • If the original number is greater than 1, the power of 10 will be 10 n .
      • If the original number is between 0 and 1, the power of 10 will be 10 n .
    4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
    1. Determine the exponent, n , on the factor 10.
    2. Move the decimal n places, adding zeros if needed.
      • If the exponent is positive, move the decimal point n places to the right.
      • If the exponent is negative, move the decimal point | n | places to the left.
    3. Check.

Practice makes perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

10 0 10 −1 + 10 −2

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2 0 2 −1 + 2 −2

3 4

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  1. ( −6 ) −2
  2. 6 −2

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  1. ( −8 ) −2
  2. 8 −2

  1. 1 64
  2. 1 64

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  1. ( −10 ) −4
  2. 10 −4

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  1. ( −4 ) −6
  2. 4 −6

  1. 1 4096
  2. 1 4096

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  1. 5 · 2 −1
  2. ( 5 · 2 ) −1

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  1. 10 · 3 −1
  2. ( 10 · 3 ) −1

  1. 10 3
  2. 1 30

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  1. 4 · 10 −3
  2. ( 4 · 10 ) −3

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  1. 3 · 5 −2
  2. ( 3 · 5 ) −2

  1. 3 25
  2. 1 225

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  1. 4 x −1
  2. ( 4 x ) −1
  3. ( −4 x ) −1

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  1. 3 q −1
  2. ( 3 q ) −1
  3. ( −3 q ) −1

  1. 3 q
  2. 1 3 q
  3. 1 3 q

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  1. 6 m −1
  2. ( 6 m ) −1
  3. ( −6 m ) −1

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  1. 10 k −1
  2. ( 10 k ) −1
  3. ( −10 k ) −1

  1. 10 k
  2. 1 10 k
  3. 1 10 k

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Simplify Expressions with Integer Exponents

In the following exercises, simplify .

x 4 · x −2 · x −3

1 x

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( x 5 y −1 ) ( x −10 y −3 )

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( a 3 b −3 ) ( a −5 b −1 )

1 a 2 b 4

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( u v −2 ) ( u −5 v −4 )

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( p q −4 ) ( p −6 q −3 )

1 p 5 q 7

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( −2 r −3 s 9 ) ( 6 r 4 s −5 )

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( −3 p −5 q 8 ) ( 7 p 2 q −3 )

21 q 5 p 3

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( −6 m −8 n −5 ) ( −9 m 4 n 2 )

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( −8 a −5 b −4 ) ( −4 a 2 b 3 )

32 a 3 b

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( 2 n −3 ) −6

n 18 64

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Convert from Decimal Notation to Scientific Notation

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

0.0000103

1.03 × 10 −5

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The population of the United States on July 4, 2010 was almost 310,000,000 .

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

6.85 × 10 9

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The average width of a human hair is 0.0018 centimeters.

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The probability of winning the 2010 Megamillions lottery is about 0.0000000057 .

5.7 × 10 −9

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Convert Scientific Notation to Decimal Form

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

1.6 × 10 10

16,000,000,000

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6.15 × 10 −8

0.0000000615

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In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was 2 × 10 4 .

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At the start of 2012, the US federal budget had a deficit of more than $1.5 × 10 13 .

$15,000,000,000,000

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The concentration of carbon dioxide in the atmosphere is 3.9 × 10 −4 .

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The width of a proton is 1 × 10 −5 of the width of an atom.

0.00001

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Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

( 2 × 10 5 ) ( 2 × 10 −9 )

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( 3 × 10 2 ) ( 1 × 10 −5 )

0.003

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( 1.6 × 10 −2 ) ( 5.2 × 10 −6 )

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( 2.1 × 10 −4 ) ( 3.5 × 10 −2 )

0.00000735

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8 × 10 6 4 × 10 −1

200,000

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7 × 10 −2 1 × 10 −8

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5 × 10 −3 1 × 10 −10

50,000,000

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Everyday math

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.5 trillion calories by the end of 2015.

  1. Write 1.5 trillion in decimal notation.
  2. Write 1.5 trillion in scientific notation.
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Length of a year The difference between the calendar year and the astronomical year is 0.000125 day.

  1. Write this number in scientific notation.
  2. How many years does it take for the difference to become 1 day?
  1. 1.25 × 10 −4
  2. 8,000
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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 probability of getting a particular 5-card hand from a deck of cards, Mario divided 1 by 2,598,960 and saw the answer 3.848 × 10 −7 . Write the number in decimal notation.

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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 6 of her 50 favorite photographs, she multiplied 50 · 49 · 48 · 47 · 46 · 45 . Her calculator gave the answer 1.1441304 × 10 10 . Write the number in decimal notation.

11,441,304,000

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Writing exercises

  1. Explain the meaning of the exponent in the expression 2 3 .
  2. Explain the meaning of the exponent in the expression 2 −3
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When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answers will vary.

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Self check

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

.

After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

Questions & Answers

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Practice Key Terms 2

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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