Chapter Seven 7.2

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§ 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
radicand becomes the base.
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
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 #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,
add exponents.
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
Add.
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
Rewrite in radical notation.
Write the radicand as an
exponential expression.
Write the entire expression in
exponential form.
To raise powers to powers,
multiply the exponents.
Rewrite in radical notation.
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
to simplify radical expressions.
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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