Math 52 (Summer ’09)
4.1 "Integers As Exponents"
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Objectives:
*
Tell the meaning of exponential notation.
*
Evaluate exponential expressions with exponents of 0 and 1.
*
Evaluate algebraic expressions containing exponents.
*
Use the product rule to multiply exponential expressions with like bases.
*
Use the quotient rule to divide exponential expressions with like bases.
*
Express an exponential expression involving negative exponents with positive exponents.
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Exponential Notation
An exponent of 2 or greater tells how many times the base is used as a factor. For example, a a a a = a4 : In this case, the
exponent is 4 and the base is a. An expression for a power is called exponential notation.
.
We read exponential notation as follows: an is read the nth power of a, or simply a to the nth, or a to the n.
There are special cases, for example we often read x2 as "x squared" and x3 as "x cubed."
De…nition:
"Natural-Number Exponents"
A natural-number exponent indicates how many times its base is to be used as a factor. For any number a (base)
and any natural number n (exponent);
.
Example 1: (Using the meaning of natural-number exponents)
What is the meaning of each of the following?
2
2
a) 43
b)
3
c)
3x4
One and Zero as Exponents
Exponents of 0 and 1 :
For any number a;
.
For any nonzero number a;
WARNING!!!
00 is not de…ned
Example 2: (Exponents of 0 and 1)
Evaluate:
0
2
a) b1
b)
3
e) ab1
.
f) a0 b
c) 8:380
0
g) (ab)
Page: 1
1
d) ( 7:03)
1
h) (ab)
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
4.1
Evaluating Algebraic Expressions
3
Algebraic expressions can involve exponential notation. For example: x4 ; (3x) 2; a2 +2ab+b2 : We evaluate algebraic
expressions by replacing variables with numbers and following the rules for order of operations.
Example 3: (Evaluating algebraic expressions)
Evaluate:
a) p1 ; when p = 19
b)
c) y 0
d) t5 + 5; when t =
5; when y = 2
x2
4; when x = 4
2
Multiplying and Dividing Powers with Like Bases
There are several rules for manipulating exponential notation to obtain equivalent expressions. We …rst consider multiplying and dividing powers with same bases.
Rule 1:
"The Product Rule"
For any number a and any positive integers m and n;
.
(i:e:When multiplying with exponential notation, if the bases are the same, keep the base and add the exponents)
Example 4: (Using the product rule)
Multiply and simplify:
a) 35 32
b) x x6
Rule 2:
c) p4 p9 p2
"The Quotient Rule"
.
For any nonzero number a and any positive integers m and n;
(i:e:When dividing with exponential notation, if the bases are the same, keep the base and subtract the exponent
of the denominator from the exponent of the numerator)
Example 5: (Using the quotient rule)
Divide and simplify:
x6
y 12
a) 2
b)
x
2y 3
3
c)
Page: 2
(2x)
3
(2x)
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
4.1
Negative Exponents
Negative Exponents
For any real number a that is nonzero and any integer n,
or
.
Example 6: (Negative exponents)
Express using positive exponents and simplify:
a)
2
3
b)
x
2
1
c)
7
t
Example 7: (Using the product and quotient rule with negative and positive exponents)
Simplify:
a) 3 5 38
b) x 2 x
c) t8 t 8
d)
m6
m12
e)
y8
y 3
f)
Page: 3
w
w
2
9
Notes by Bibiana Lopez