How Do We Use Rational
Exponents?
• Do Now: Perform the indicated operation
and simplify
1.
2.
1
nth Roots
nth Roots
An nth root of number a is a number whose nth
power is a.
n
a  a number whose nth power is a
If the index n is even, then the radicand a must
be nonnegative.
4
16  2, but 4 16 is not a real number
5
32  2
2
age 393
Square Root of x2
x x
2
3
Radicals
4
Rational Exponents
5
Exponent 1/n When n Is Even
6
When n Is Even
1
2
100  100  10
1
4
625  625  5
4
1
6
64  64  2
 4
6
1
2
  4 is not yet defined
7
Exponent 1/n When n Is Odd
8
Exponent 1/n When n Is Odd
1
3
27  27  3
3
 27 
1
3
  27  3
3
1
5
1
1
 1 

  5
32 2
 32 
9
nth Root of Zero
0 0
n
10
Rational Exponents
11
Evaluating in Either Order
8
 8
 2  4
2
3

2
3
 8  64  4
2
3
2
or
8
3
2
3
12
Negative Rational Exponents
13
Evaluating a-m/n
8
2

3

1
8
2
3

1
 8
3
2
1
1
 2 
2 4
14
Rules for Rational Exponents
15
7
Simplifying
y 
1
6 6

6
y  y
6


 a b ab  




1
2
1

3
16
Simplifying
y 
1
6 6

6
y  y
6



 1 1
 a b ab    a b  a b








1
2
1

3
1
2
a
1

3
 
1
1
1  1
3
2
b
3
2
a b
2
3
17
Simplifying
y 
1
6 6

6
y  y
6


 a b ab   a b




1
2
1

3
9 x y
8
3
2

1
10 12 2
z

2
3
18
Multiplying Radicals – Different Indices
1
4
1
2
4
2  2  2 2  2
3
2 3 
1 1

4 2
3
4
2  2  8
4
3
4
19
Multiplying Radicals
Different Indices
4
3
1
4
1
2
1
3
1
2
2  2  2 2  2
1 1

4 2
3
4
2  2  8
4
3
4
2  3  2 3 
20
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2  8
4
3
4
3
6
2  3  2 3  2 3 
21
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2  8
3
6
3
4
4
2  3  2 3  2 3  2  3 
6
2
6
3
22
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2 4 8
3
6
3
4
2  3  2  3  2  3  2  3  108
6
2
6
3
6
23
Rational Exponents
Eliminate the root, then the power
2
3
a 2
24
Eliminate the Root, Then the Power
2
3
a 2
3
 
 a   23
 
 
2
a 8
2
3
a  8
2
a  2 2
CHECK
25
Negative Exponents
r  1
2

3
1
26
Negative Exponents
Eliminate the root, then the power
r  1

2
3
1
3
 r  1   13




2

3
r  1  1
2
r  1  
2
r  1  1
r2
CHECK
1
r 0
27
Negative Exponents
Eliminate the root, then the power
2t  3

2
3
 1
28
No Solution
Eliminate the root, then the power
2t  3

2
3
 1
3
 2t  3    13




2

3
2t  3  1
2
2t  3  
2
1
29
No Solution
Eliminate the root, then the power
2t  3

2
3
 1
3
 2t  3    13




2

3
2t  3  1
2
2t  3  
2
1
No real solution
30
Strategy for Solving Equations with
Exponents and Radicals
31
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