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10.5 Integer exponents and scientific notation  (Page 4/7)

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Write in scientific notation: 0.0052 .

Solution

0.0052
Move the decimal point to get 5.2, a number between 1 and 10. .
Count the number of decimal places the point was moved. 3 places
Write as a product with a power of 10. 5.2 × 10 −3

Check your answer:

5.2 × 10 −3 5.2 × 1 10 3 5.2 × 1 1000 5.2 × 0.001 0.0052 0.0052 = 5.2 × 10 −3

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Write in scientific notation: 0.0078 .

7.8 × 10 −3

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Write in scientific notation: 0.0129 .

1.29 × 10 −2

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Convert scientific notation to decimal form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

9.12 × 10 4 9.12 × 10 −4 9.12 × 10,000 9.12 × 0.0001 91,200 0.000912

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Convert to decimal form: 6.2 × 10 3 .

Solution

Step 1 : Determine the exponent, n , on the factor 10. 6 × 10 3
Step 2 : Move the decimal point 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.
 6,200
Step 3 : Check to see if your answer makes sense.
10 3 is 1000 and 1000 times 6.2 will be 6,200.
6.2 × 10 3 = 6,200
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Convert to decimal form: 1.3 × 10 3 .

1,300

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Convert to decimal form: 9.25 × 10 4 .

92,500

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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.

Convert to decimal form: 8.9 × 10 −2 .

Solution

8.9 × 10 −2
Determine the exponent n , on the factor 10. The exponent is −2.
Move the decimal point 2 places to the left. .
Add zeros as needed for placeholders. 0.089
8.9 × 10 −2 = 0.089
The Check is left to you.
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Convert to decimal form: 1.2 × 10 −4 .

0.00012

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Convert to decimal form: 7.5 × 10 −2 .

0.075

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Multiply and divide using scientific notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Multiply. Write answers in decimal form: ( 4 × 10 5 ) ( 2 × 10 −7 ) .

Solution

( 4 × 10 5 ) ( 2 × 10 −7 )
Use the Commutative Property to rearrange the factors. 4 · 2 · 10 5 · 10 −7
Multiply 4 by 2 and use the Product Property to multiply 10 5 by 10 −7 . 8 × 10 −2
Change to decimal form by moving the decimal two places left. 0.08
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Multiply. Write answers in decimal form: ( 3 × 10 6 ) ( 2 × 10 −8 ) .

0.06

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Multiply. Write answers in decimal form: ( 3 × 10 −2 ) ( 3 × 10 −1 ) .

0.009

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Divide. Write answers in decimal form: 9 × 10 3 3 × 10 −2 .

Solution

9 × 10 3 3 × 10 −2
Separate the factors. 9 3 × 10 3 10 −2
Divide 9 by 3 and use the Quotient Property to divide 10 3 by 10 −2 . 3 × 10 5
Change to decimal form by moving the decimal five places right. 300,000
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Divide. Write answers in decimal form: 8 × 10 4 2 × 10 −1 .

400,000

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Divide. Write answers in decimal form: 8 × 10 2 4 × 10 −2 .

20,000

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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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