Transcript Beginning & Intermediate Algebra, 4ed

Chapter 5
Exponents and
Polynomials
§ 5.1
Exponents
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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Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions.
34 = 3 • 3 • 3 • 3 = 81
(–5)2 = (– 5)(–5) = 25
–62 = – (6)(6) = –36
(2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512
3 • 42 = 3 • 4 • 4 = 48
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Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions.
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
Product Rule for Exponents
If m and n are positive integers and a is a real number,
then
am · an = am+n
Example: Simplify each of the following expressions.
32 · 34 = 32+4 = 36 = 3 · 3 · 3 · 3 · 3 · 3= 729
x4 · x5 = x4+5 = x9
z3 · z2 · z5= z3+2+5 = z10
(3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6
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The Power Rule
Power Rule for Exponents
If m and n are positive integers and a is a real number,
then
(am)n = amn
Example:
Simplify each of the following expressions.
(23)3 = 23·3 = 29 = 512
(x4)2 = x4·2 = x8
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The Power of a Product Rule
Power of a Product Rule
If n is a positive integer and a and b are real numbers,
then
(ab)n = an · bn
Example:
Simplify (5x2y)3.
(5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3
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The Power of a Quotient Rule
Power of a Quotient Rule
If n is a positive integer and a and c are real numbers,
then
n
a
an
 n, c0
c
c

Example:
3
 3x z 
Simplify 
 .
2
y


2
3
 3x z  33(x 2)3 z 3 27 x 6 z 3
 2 y   23 y 3  8 y 3


2
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The Quotient Rule
Quotient Rule for Exponents
If m and n are positive integers and a is a real number,
then a m
mn

a
an
as long as a is not 0.
Example:
Simplify the following expression.
9a 4b 7   9  a 4  b 7 
41
7 2
3 5



3
(
a
)(
b
)

3
a
b
2 



2
 3  a  b 
3ab
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Zero Exponent
Zero Exponent
a0 = 1, as long as a is not 0.
Example:
Simplify each of the following expressions.
50 = 1
(xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1
–x0 = –(x0) = – 1
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