# Chapter Seven 7.2

```§ 7.2
Rational Exponents
Rational Exponents
The Definition of
If
n
a
represents a real number and
a
1/ n

n
n 2
a
T
1/ n
is an integer, then
a.
If a is negative, n must be odd. If a is nonnegative, n can be
any index.
P 499
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2
Rational Exponents
EXAMPLE
Use radical notation to rewrite each expression. Simplify, if
possible:
1
1
1
4
(a) 3 xy 5 (b) 100 2 (c)   64  3 .
SOLUTION

(a) 3 xy
4

1
5

5
3 xy
4
1
(b) 100
2

100  10
1
(c)   64  3 
3
 64   4
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.2
Rational Exponents
Check Point 1 on page 499
1

(a) 25 2
1
(b)   8  3

(c) 5 xy
2


1
4

25  5
 8  2
3
4
5 xy
2
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.2
Rational Exponents
EXAMPLE
Rewrite with rational exponents:
(a)
5
13 x
5
(b)
x .
SOLUTION
Parentheses are needed in part (a) to show that the entire
1
(a)
(b)
5
13 x  13 x  5
 
x  x
5
5
1
2
 x
5
1
5
2
 x2
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.2
Rational Exponents
Check Point 2 on p 500
1
(a)
4
5 xy
 5 xy  4
1
3
(b)
5
a b
2
 a b 5

 

2


3
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.2
Rational Exponents
The Definition of
m
a
T
m/n
If a represents a real number, is a positive rational
n
number reduced to lowest terms, and n  2 is an integer, then
n
a
m/n

a
m/n

and
 a
m
n
n
m
a .
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.2
Rational Exponents
EXAMPLE
Use radical notation to rewrite each expression and simplify:
3
2
(b)   27  3 .
(a) 25 2
SOLUTION
3
(a) 25
2


2

3
25
(b)   27  3 

3
 5  125
 27
3

2
  3   9
2
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.2
Rational Exponents
Check Point 3 on p 501
4
(a) 8  3

 8
3
(c) - 81
4

 16
4
3

4

3
81
  27
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.2
Rational Exponents
EXAMPLE
Rewrite with rational exponents:
(a)
7
x
4

(b)

3
11 xy .
SOLUTION
4
(a)
(b)
7

x
4
 x7

3
11 xy
3
 11 xy  2
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.2
Rational Exponents
Check Point 4 on p 501
4
3
(a)
(b)

5
6
 63
4

7
2 xy
7
  2 xy  5
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.2
Rational Exponents
The Definition of
If
a
m/n
is a nonzero real number, then
a
m / n
1

a
m/n
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.2
.
a
m /n
T
Rational Exponents
Check Point 5 on p 502

1

2
(a) 100
1
100

(b) 8
3

5
(d) 3 xy 
1
83
2
1
1

8
3
5
9

1
32

10
2
1
3

(c) 32

1

1
1

5
1
5
3 xy  9
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.2
Rational Exponents in Application
EXAMPLE
The Galapagos Islands, lying 600 miles west of Ecuador, are
famed for their extraordinary wildlife. The function
1
f  x   29 x 3
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?
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.2
Rational Exponents in Application
CONTINUED
SOLUTION
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).
1
f  x   29 x 3
This is the given formula.
1
f  27   29  27  3
f  27   29
Replace x with 27.
1
3
27
Rewrite  27  3 as
3
27
.
f  27   29  3
Evaluate the cube root.
f  27   87
Multiply.
A Galapagos island having an area of 27 square miles contains
approximately 87 plant species.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.2
Rational Exponents
p 502
Properties of Rational Exponents
If m and n are rational exponents, and a and b are real numbers for
which the following expressions are defined, then
1)
b b  b
m
n
2)
b
m
b
n
b
mn
When multiplying exponential expressions with
the same base, add the exponents. Use this sum
as the exponent of the common base.
mn
When dividing exponential expressions with the
same base, subtract the exponents. Use this
difference as the exponent of the common base.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.2
Rational Exponents
p 502
CONTINUED
Properties of Rational Exponents
If m and n are rational exponents, and a and b are real numbers for
which the following expressions are defined, then
3)
b 
b
4)
ab n
a b
5)
a
a
   n
b
b
m n
n
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.
mn
n
n
n
When a product (not sum) is raised to a power,
raise each factor to that power and multiply.
When a quotient is raised to a power, raise the
numerator to that power and divide by the
denominator to that power.
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.2
Rational Exponents
EXAMPLE
1
3
Simplify:
(a)
x7
1
x
7


4
(b)  x y 5


1
2
3



3
(c)
1
54 52
1
.
54
SOLUTION
3
(a)
x7
1
x
3
 x7
7
2
 x7

1
7
To divide with the same base,
subtract exponents.
Subtract.
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.2
Rational Exponents
CONTINUED
1
 1 2
(b)  x 4 y 5


1
1
3  1
  x4




3  2
 y 5
 
 
1
 x
12

y
3



2
To raise a product to a power, raise
each factor to the power.
Multiply: 1  1
2 1
2

and    
.
4 3 12
5 3
15
15
1
1

x 12
Rewrite with positive exponents.
2
y 15
3
(c)
1
3
54 52
54
1
54


1
54
1
3
2
54


1
54
2
5
4
54

1
5
 54

1
4
4
 54  5  5
54
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.2
1
Rational Exponents
Check Point 6 on page 503
1
1
(a) 7 2 7 3
 72
1

1
3
3
 76

Simplify:
2
5
6
 76
1
(b)
1
50 x 3
 5x3
4
10 x

4
3
 5x

1
x
3
3

(c)  9 . 1 5


2




4
6

  9 . 1 20






5
3
 9 . 1 10
1
1
(d)
 3 1
x 5y4


3



 3 1
  x 15 y 12







y 12
1
x5
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.2
Rational Exponents
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 #21 Section 7.2
Rational Exponents
EXAMPLE
Use rational exponents to simplify: (a)
6
ab 
2
3
2
a b
(b)
5 3
2x .
SOLUTION
(a)
6

3
2
 a b 
1
ab  a b  ab
2
1
2
6
1
1
2
2
2
6
2
1
 a 6b 6 a 3b 3
1
4
1
1
 a 6 a 6b 3b 3
1
 a6
3
  a 
 a6 b

4
6
b
Rewrite as exponential
expressions.
1
2
1 1

3 3
1
1
3
b3
Raise each factor in parentheses
to its related power.
To raise powers to powers,
multiply.
Reorder the factors.
To multiply with the same base,
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.2
Rational Exponents
5
CONTINUED
2
 a 6b 3
5
4
a b
6

 a b
(b)
5 3

6
2x 
5
5
Rewrite exponents with
common denominators.
6
4
5
a b

1
Factor 1/6 out of the exponents.
6
4
1
2 x  3
1
1 5


  2 x  3 


1
  2 x 15

15
2x
exponential expression.
Write the entire expression in
exponential form.
To raise powers to powers,
multiply the exponents.
Blitzer, Intermediate Algebra, 5e – Slide #23 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.
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 7.2
DONE
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
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 7.2
Rational Exponents
Important to Remember:
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 7.2
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