Download Exponents

Chapter 12
Exponents and
Polynomials
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
12.1
Exponents
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Exponents
Exponents that are natural numbers are
shorthand notation for repeating factors.
34 = 3 • 3 • 3 • 3
3 is the base
4 is the exponent (also called power)
Note by the order of operations that exponents
are calculated before other operations.
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Example
Evaluate each expression.
a. 34 = 3 • 3 • 3 • 3 = 81
b. (–5)2 = (– 5)(–5) = 25
c. –62 = – (6)(6) = –36
d. (2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512
e. 3 • 42 = 3 • 16 = 48
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Example
Evaluate each expressions for the given value of x.
a. Find 3x2 when x = 5.
3x2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75
b. Find –2x2 when x = –1.
–2x2 = –2(–1)2 = –2(–1)(–1) = –2(1) = –2
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The Product Rule for Exponents
If m and n are positive integers and a is a real
number, then
am · an = am+n
Example:
a. 32 · 34
= 32+4 = 36
b. x4 · x5 = x4+5 = x9
c. z3 · z2 · z5
= z3+2+5 = z10
d. (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6
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Helpful Hint
Don’t forget that
35 ∙ 37 = 912
Add exponents.
Common base not kept.
35 ∙ 37 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
5 factors of 3.
7 factors of 3.
= 312 12 factors of 3, not 9.
In other words, to multiply two exponential expressions
with the same base, we keep the base and add the
exponents. We call this simplifying the exponential
expression.
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Helpful Hint
Don’t forget that if no exponent is written, it is
assumed to be 1.
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The Power Rule
If m and n are positive integers and a is a real
number, then
(am)n = amn
Example:
a. (23)3 = 23·3 = 29
b. (x4)2 = x4·2 = x8
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Power of a Product Rule
If n is a positive integer and a and b are real
numbers, then
(ab)n = an · bn
Example:
(5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3
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Power of a Quotient Rule
If n is a positive integer and a and c are real numbers, then
n
 a
an
 c   n ,c  0
c
Example:
4
 p
p4
 4   4
4
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Quotient Rule for Exponents
If m and n are positive integers and a is a real
number, then
m
a
mn

a
,a  0
n
a
Example:
4 7
9a b
3ab 2
4
7




9
a
b
 
    2   3(a 41 )(b72 )  3a 3b 5
 3  a  b 
Group common
bases together.
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Zero Exponent
a0 = 1, as long as a is not 0.
Note: 00 is undefined.
Example:
a. 50 = 1
b. (xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1
c. –x0 = –(x0) = – 1
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