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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter 12
Exponents and Polynomials
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
12.1
Exponents
Martin-Gay, Developmental Mathematics, 2e 33
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.
Martin-Gay, Developmental Mathematics, 2e 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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
Example
Martin-Gay, Developmental Mathematics, 2e 55
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Evaluate each expressions for the given value of x.
Example
a. Find 3x2 when x = 5.
b. Find –2x2 when x = –1.
3x2 = 3(5)2 = 3(5 · 5) = 3 · 25
–2x2 = –2(–1)2 = –2(–1)(–1) = –2(1)
= 75
= –2
Martin-Gay, Developmental Mathematics, 2e 66
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
If m and n are positive integers and a is a real number, then
am · an = am+n
a. 32 · 34 = 36
b. x4 · x5 = x4+5
c. z3 · z2 · z5 = z3+2+5
d. (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6
= 32+4
= x9
= z10
The Product Rule for Exponents
Example:
Martin-Gay, Developmental Mathematics, 2e 77
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Helpful Hint
Don’t forget that
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.
35 ∙ 37 = 912
35 ∙ 37 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
= 312 12 factors of 3, not 9.
Add exponents.
Common base not kept.
5 factors of 3. 7 factors of 3.
Martin-Gay, Developmental Mathematics, 2e 88
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Helpful Hint
Don’t forget that if no exponent is written, it is assumed to be 1.
Martin-Gay, Developmental Mathematics, 2e 99
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
If m and n are positive integers and a is a real number, then
(am)n = amn
Example:
a. (23)3 = 29
b. (x4)2 = x8
= 23·3
= x4·2
The Power Rule
Martin-Gay, Developmental Mathematics, 2e 1010
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
If n is a positive integer and a and b are real numbers, then
(ab)n = an · bn
Power of a Product Rule
Example:
= 53 · (x2)3 · y3 = 125x6 y3(5x2y)3
Martin-Gay, Developmental Mathematics, 2e 1111
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
If n is a positive integer and a and c are real numbers, then
Power of a Quotient Rule
ac
n
an
cn,c 0
Example:
p4
4
p4
44
Martin-Gay, Developmental Mathematics, 2e 1212
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Quotient Rule for Exponents
Example:
If m and n are positive integers and a is a real number, then
am
anam n,a 0
2
74
3
9
ab
ba 533 ba))((3 2714 ba
2
74
3
9
b
b
a
a
Group common bases together.
Martin-Gay, Developmental Mathematics, 2e 1313
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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
Zero Exponent
