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§ 7.2
Rational Exponents
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2
Rational Exponents
The Definition of TIf represents a real number and is an integer, then
If a is negative, n must be odd. If a is nonnegative, n can be any index.
n a
na /1
2n
./1 nn aa
P 499
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.2
Rational ExponentsEXAMPLE
Use radical notation to rewrite each expression. Simplify, if possible:
.64(c)100(b)3(a) 31
21
51
4 xy
SOLUTION
5 451
4 33(a) xyxy
10100100(b) 21
46464(c) 331
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.2
Rational ExponentsCheck Point 1 on page 499
21
25(a)
31
8(b)
525
283
41
25(c) xy 4 25xy
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.2
Rational ExponentsEXAMPLE
Rewrite with rational exponents:
.(b)13(a) 55 xx
SOLUTION
51
5 1313(a) xx
Parentheses are needed in part (a) to show that the entire radicand becomes the base.
25
215
21
55(b) xxxx
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.2
Rational ExponentsCheck Point 2 on p 500
4 5(a) xy 41
5xy
53
2(b) ba 5
13
2
ba
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.2
Rational Exponents
The Definition of TIf represents a real number, is a positive rational number reduced to lowest terms, and is an integer, then
and
n a
nma /
2n
mnnm aa /
./ n mnm aa
nm
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.2
Rational ExponentsEXAMPLE
Use radical notation to rewrite each expression and simplify:
.27(b)25(a) 32
23
SOLUTION
12552525(a) 3323
932727(b) 22332
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.2
Rational ExponentsCheck Point 3 on p 501
43
81-(c)
34
8(a) 43 8 16
34 81 27
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.2
Rational ExponentsEXAMPLE
Rewrite with rational exponents:
.11(b)(a)37 4 xyx
SOLUTION
74
7 4(a) xx
233
1111(b) xyxy
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.2
Rational ExponentsCheck Point 4 on p 501
3 46(a) 34
6
75 2(b) xy 57
2xy
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.2
Rational Exponents
The Definition of TIf is a nonzero real number, then
nma /
nma /
.1/
/nm
nm
aa
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.2
Rational ExponentsCheck Point 5 on p 502
21
100(a)
101
21
100
1
31
8(b)
21
31
8
1
53
32(c)
81
53
32
1
95
3(d) xy 9
53
1
xy
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.2
Rational Exponents in ApplicationEXAMPLE
The Galapagos Islands, lying 600 miles west of Ecuador, are famed for their extraordinary wildlife. The function
models the number of plant species, f (x), on the various islands of the Galapagos chain in terms of the area, x, in square miles, of a particular island. Use the function to solve the following problem.
How many species of plants are on a Galapagos island that has an area of 27 square miles?
31
29xxf
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.2
Rational Exponents in Application
Because we are interested in how many species of plants there are on a Galapagos island having an area of 27 square miles, substitute 27 for x. Then calculate f (x).
SOLUTION
31
29xxf
CONTINUED
This is the given formula.
31
272927 f Replace x with 27.
3 272927 f Rewrite as .
32927 f Evaluate the cube root.
31
27 3 27
8727 f Multiply.A Galapagos island having an area of 27 square miles contains approximately 87 plant species.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.2
Rational Exponents p 502
Properties of Rational ExponentsIf m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then
1) When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.
2) When dividing exponential expressions with the same base, subtract the exponents. Use this difference as the exponent of the common base.
nmnm bbb
nmn
m
bbb
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.2
Rational Exponents p 502
Properties of Rational ExponentsIf m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then
3) When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.
4) When a product (not sum) is raised to a power, raise each factor to that power and multiply.
5) When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.
CONTINUED
mnnm bb
nnn baab
n
nn
ba
ba
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.2
Rational ExponentsEXAMPLE
Simplify: .5
55(c)(b)(a)41
21
43
31
52
417
3
71
yx
x
x
SOLUTION
72
71
737
3
71(a)
x
xx
x
To divide with the same base,
subtract exponents.
Subtract.
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.2
Rational Exponents
152
121
152
121
31
523
1
413
1
52
41
(b)
y
x
yx
yxyx
To raise a product to a power, raise each factor to the power.
Multiply:
CONTINUED
55555
5
5
5
5
5
5
55(c) 144
41
45
41
45
41
42
43
41
21
43
41
21
43
.152
31
52 and
121
31
41
Rewrite with positive exponents.
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.2
Rational ExponentsCheck Point 6 on page 503Simplify:
43
52
1.9(c)
34
10
50(b)31
x
x
31
21
77(a) 31
21
7
62
63
7
65
7
34
31
5
x15 x x
5
20
6
1.9 103
1.9
31
41
53
(d)
yx
121
153
yx51
121
x
y
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.2
Rational Exponents
Simplifying Radical Expressions Using Rational Exponents
1) Rewrite each radical expression as an exponential expression with a rational exponent.2) Simplify using properties of rational exponents.3) Rewrite in radical notation if rational exponents still appear.
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.2
Rational ExponentsEXAMPLE
Use rational exponents to simplify: .2(b)(a) 5 33 26 2 xbaab
SOLUTIONRewrite as exponential expressions.Raise each factor in parentheses to its related power.
31
261
23 26 2(a) baabbaab
31
31
261
261
baba
31
32
62
61
baba
31
31
64
61
bbaa
31
31
64
61
ba
To raise powers to powers, multiply.Reorder the factors.
To multiply with the same base, add exponents.
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 7.2
Rational Exponents
531
5 3 22(b) xx
Add.32
65
ba
Rewrite exponents with common denominators.
64
65
ba
Factor 1/6 out of the exponents. 61
45ba
Rewrite in radical notation.6 45ba
51
31
2
x
151
2x
Write the radicand as an exponential expression.Write the entire expression in exponential form.To raise powers to powers, multiply the exponents.
15 2x Rewrite in radical notation.
CONTINUED
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 7.2
Rational Exponents
Important to Remember:
• An expression with rational exponents is simplified when no parentheses appear,
no powers are raised to powers, each base occurs once, and
no negative or zero exponents appear.
• Some radical expressions can be simplified using rational exponents. Rewrite the expression using rational exponents, simplify, and rewrite in radical notation if rational exponents still appear.
DONE
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 7.2
Rational Exponents
Rational exponents have been defined in such a way so as to make their properties the same as the properties for integer exponents.
In this section we explore the meaning of a base raised to a rational (fractional) exponent.
We will also discover how we can use rational exponents to simplify radical expressions.
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 7.2
Rational Exponents
Important to Remember: