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Section 5.2

Negative Exponents and

Scientific Notation

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Negative Exponents

Definition of a Negative Exponent

where x 01nn

xx

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Write with positive exponents.

a.4h

41h

b. 532a

5 153

1 1322

aa

Example

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Negative Exponents

Laws of Exponents Where x, y, ≠ 0

The Product Rule

The Quotient Rule

Power Rules

a b a bx x x

aa b

b

xx

x Use if a > b. 1a

b b a

x

x x Use if a < b.

, , a a

ba a a a aba

x xxy x y x x

y y

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Negative Exponents

Properties of Negative Exponents Where x, y, ≠ 0

1 nn

xx

m n

n m

x y

y x

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Example

Simplify. Write the expression with no negative exponents.

a.3 5

2

x y

x y

3 2 5

5 6 x x xyy y

b. 34 22ab c 3 3 ( 4)( 3) (2)( 3)2 a b c

3 3 12 62 a b c12

3 3 62 b

a c

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Scientific Notation

Scientific Notation A positive number is written in scientific notation if it is in the form a × 10n, where 1 a 10 and n is an integer.

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Scientific Notation

8200 = 8.2 1000 = 8.2 103

34,200,000 = 3.42 10000000 = 3.42 107

Greater than 1 and less than 10

Power of 10

Scientific notation

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Write 67,300 in scientific notation.

67,300. = 6.73 10n

The decimal point was moved 4 places to the left, so we use a power of 4.

67,300 = 6.73 104

A number that is larger than 10 and written in scientific notation will always have a positive exponent as the power of 10.

Example

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Write 0.048 in scientific notation.

0.048 = 4.8 10n

The decimal point was moved 2 places to the right, so we use a power of –2.

0.048 = 4.8 10–2

A number that is smaller than 1 and written in scientific notation will always have a negative exponent as the power of 10.

Example

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Write 9.1 104 in decimal notation.

9.1 104 = 9.1000 104= 91,000

Write 6.72 10–3 in decimal notation.

6.72 10–3= 6.72 10–3= 0.00672

Example

Move the decimal point 4 places to the right.

Move the decimal point 3 places to the left.

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Example

Use scientific notation and the laws of exponents to find the following. Leave your answer in scientific notation.

32,000,000 1,500,000,000,000

7 123.2 10 1.5 10

7 123.2 1.5 10 10

194.8 10

Write each number in scientific notation.

Rearrange the order.

Multiply.

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Example

Use scientific notation and the laws of exponents to find the following. Leave your answer in scientific notation.

0.00063

0.021

4

2

6.3 10

2.1 10

4

2

6.3 10

2.1 10

2

4

6.3 10

2.1 10

23.0 10

Write each number in scientific notation.

Rearrange the order.

Rewrite with positive exponents.