Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.6 Rules of Exponents and Scientific Notation.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 5.6

Rules of Exponents

and Scientific Notation


What You Will Learn

ExponentsRules of ExponentsScientific Notation

5.6-2


Exponents

Given 52, 2 is the exponent, 5 is the baseRead 52 as “5 to the second power” or “5 squared,” which means

5.6-3


Exponents“5 to the third power,” or “5 cubed” is

“b to the nth power,” or bn, meansmulitiply b n times.

5.6-4


Example 1: Evaluating the Power of a NumberEvaluate.a) 52

= 5 • 5= 25

b) (–3)2 = (–3) • (–3)

= 9

c) 34 = 3 • 3 • 3 •

3= 81

d) 11000 = 1

e) 10001 = 10005.6-5


Example 2: The Importance of ParenthesesEvaluate.a) (–2)4 = (–2)(–2)(–2)(–

2)= 4(–2)(–2)

b) –24 = –1 • 24 = –1 • 2 • 2 • 2 •

2

=–8(–2)

= 16

= –1 • 16= –16

5.6-6


Product Rule for Exponents

am an amn

5.6-7


Example 3: Using the Product Rule for ExponentsUse the product rule to simplify.

a) 33 • 32

= 33+2 = 35

b) 5 • 53 = 51 • 53 = 51+3

=243

= 54 = 625

5.6-8


Quotient Rule for Exponents

am

anam n, a 0

5.6-9


Example 4: Using the Quotient Rule for ExponentsUse the quotient rule to simplify.

= 37–5 = 32

= 59–5

=9

= 54 = 625

a)

37

35

b)

59

55

5.6-10


Zero Exponent Rule

a0 1, a 0

5.6-11


Example 5: The Zero PowerUse the zero exponent rule to simplify. Assume x ≠ 0.a) 20

= 1

b) (–2)0

c) –20 = –1 • 20= –1 • 1

= 1

= –1

d) (5x)0

= 1

e) 5x0 = 5 • x0 = 5 • 1= 5 5.6-12


Negative Exponent Rule

a m

1

am, a 0

5.6-13


Example 6: Using the Negative Exponent RuleUse the negative exponent rule to simplify.

a) 2 3

1

23

1

8

b) 5 1

1

51

1

5

5.6-14


Power Rule for Exponents

am n amn

5.6-15


Example 7: Evaluating a Power Raised to Another PowerUse the power rule to simplify.

a) (54)3

= 512 = 54•3

b) (72)5

= 710 = 72•5

5.6-16


Rules for Exponents

am n amn

am

anam n, a 0

a0 1, a 0

a m

1

am, a 0

am an amn Product Rule

Quotient Rule

Zero Exponent Rule

Negative Exponent Rule

Power Rule

5.6-17


Scientific Notation

Many scientific problems deal with very large or very small numbers.Distance from the Earth to the sun is 93,000,000,000,000 miles.Wavelength of a yellow color of light is 0.0000006 meter.

5.6-18


Scientific notation is a shorthand method used to write these numbers.

93,000,000 = 9.3 × 10,000,000 = 9.3 × 107

0.0000006 = 6.0 × 0.0000001

= 6.0 × 10–7

Scientific Notation

5.6-19


Example 8: Converting from Decimal Notation to Scientific NotationWrite each number in scientific notation.a) In 2010, the population of the United State was about 309,500,000.

309,500,000 = 3.095 × 108

b) In 2010, the population of the China was about 1,348,000,000.

1,348,000,000 = 1.348 × 109

5.6-20


Example 8: Converting from Decimal Notation to Scientific Notationc) In 2010, the population of the

world was about 6,828,000,000.6,828,000,000 = 6.828 × 109

d) The diameter of a hydrogen atom nucleus is about 0.0000000000011 millimeter.0.0000000000011 = 1.1 × 10–12

5.6-21


Example 8: Converting from Decimal Notation to Scientific Notatione) The wavelength of an x-ray is

about 0.000000000492.0.000000000492 = 4.92 × 10–10

5.6-22


Example 9: Converting from Scientific Notation to Decimal NotationWrite each number in decimal notation.a) The average distance from Mars to

the sun is about 1.4 × 108 miles.1.4 × 108 = 140,000,000

b) The half-life of uranium-235 is about 4.5 × 109 years.4.5 × 109 = 4,500,000,000

5.6-23


Example 9: Converting from Scientific Notation to Decimal Notationc) The average grain size in

siltstone is about 1.35 × 10–3 inch.1.35 × 10–3 = 0.00135

d) A millimicron is a unit of measure used for very small distances. One millimicron is about 3.94 × 10–8 inch.3.94 × 10–8 = 0.0000000394

5.6-24


Example 10: Multiplying Numbers in Scientific NotationMultiply (2.1 × 105)(9 × 10–3). Write the answer in scientific notation and in decimal notation.(2.1 × 105)(9 × 10–3)

= (2.1 × 9)(105 × 10–3)= 18.9 × 102

= 1890

5.6-25


Example 11: Dividing Numbers in Scientific Notation

Divide .

Write the answer in scientific notation and in decimal notation.

0.000000000048

24,000,000,000

4.8 10 11

2.4 1010

0.000000000048

24,000,000,000

4.8 10 11

2.4 1010

5.6-26


Example 11: Dividing Numbers in Scientific NotationSolution

0.000000000048

24,000,000,000

4.8 10 11

2.4 1010

4.8

2.4

10 11

1010

2.0 10 1110

2.0 10 21

5.6-27

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.6 Rules of Exponents and Scientific Notation.

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