Download Prealgebra & Introductory Algebra

Chapter 3
Exponents and
Polynomials
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
3.1
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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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
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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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
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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The Product Rule
If m and n are positive integers and a is a real
number, then
am · an = am+n
For example,
32 · 34 = 32+4 = 36
x4 · x5 = x4+5 = x9
z3 · z2 · z5= z3+2+5 = z10
(3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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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.
= 312
7 factors of 3.
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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Helpful Hint
Don’t forget that if no exponent is written, it is
assumed to be 1.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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The Power Rule
If m and n are positive integers and a is a real
number, then
(am)n = amn
For example,
(23)3 = 23·3 = 29
(x4)2 = x4·2 = x8
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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The Power of a Product Rule
If n is a positive integer and a and b are real numbers, then
(ab)n = an · bn
For example,
(5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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The 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
For example,
4
 p
p4
 4   4
4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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The Quotient Rule
If m and n are positive integers and a is a real number, then
am
mn
 a ,a  0
n
a
For 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.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Zero Exponent
a0 = 1, a ≠ 0
Note: 00 is undefined.
For example,
50 = 1
(xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1
–x0 = –(x0) = – 1
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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