copyright 2013, 2010, 2007, pearson, education, inc. section 5.6 rules of exponents and scientific...
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 5.6
Rules of Exponents
and Scientific Notation
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
ExponentsRules of ExponentsScientific Notation
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Exponents
Given 52, 2 is the exponent, 5 is the baseRead 52 as “5 to the second power” or “5 squared,” which means
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Exponents“5 to the third power,” or “5 cubed” is
“b to the nth power,” or bn, meansmulitiply b n times.
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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
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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
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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
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Quotient Rule for Exponents
am
anam n, a 0
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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