1. The document discusses radical expressions and rational exponents. It defines radicals, radical expressions, nth roots, and rational exponents. 2. Examples are provided for simplifying radical expressions using properties of exponents, including the product rule and quotient rule for square roots. 3. Techniques are described for adding, subtracting, and multiplying radical expressions by simplifying radicals before combining terms.

1. Unit 3B
Radical Expressions and
Rational Exponents

2. Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
This symbol is the radical or the radical sign
n
a
index
radical sign
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.

3. Radical Expressions
The symbol represents the negative root of a number.
The above symbol represents the positive or principal
root of a number.

Radicals

4. Square Roots
If a is a positive number, then
a is the positive square root of a and
100 
a
 is the negative square root of a.
A square root of any positive number has two roots – one is
positive and the other is negative.
Examples:
10
25
49

5
7
0.81  0.9
36
  6

9
  non-real #

8
x 4
x
Radicals

5. Rdicals
Cube Roots
3
27 
A cube root of any positive number is positive.
Examples:
3
5
4
3
125
64

3
8
  2

A cube root of any negative number is negative.
3
a

3 3
x x 
3 12
x
4
x
Radicals

6. nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
2
4
81  3
4
16 
5
32
  2

4
3  81
4
2  16
 
5
2
  32

Radicals

7. nth Roots
4
16
 
An nth root of any number a is a number whose nth power is a.
Examples:
1

5
1
 
Non-real number
6
1
  Non-real number
3
27
  3

Radicals

9. Rational Exponents
The value of the numerator represents the power of the
radicand.
Examples:
:
n
m
a
of
Definition
The value of the denominator represents the index or root of
the expression.
n m
a or  m
n
a
3
1
27
25
2
1
25 3
5 3
27
  7
2
1
2 
x
3
4
2
3
4 64
 
7 2
1
2 
x
8

10. Rational Exponents
More Examples:
:
n
m
a
of
Definition n m
a or  m
n
a
3
2
3
2
27
1
3
2
27
1






3 2
3 2
27
1
9
1
3
3
729
1
3
2
3
2
27
1
3
2
27
1





  
 2
3
2
3
27
1
9
1
 
 2
2
3
1
or

11. Rational Exponents
Examples:
:
n
m
a
of
Definition

n m
a
1
 m
n
a
1
2
1
25
1
2
1
25

25
1
5
1
3
2
1
x
3
2

x 3 2
1
x  2
3
1
x
n
m
a
1
or or
or

12. Rational Exponents
Use the properties of exponents to simplify each expression
3
5
3
4
x
x  3
9
x
3
x
10
1
5
3 
x
10
1
5
3
x
x
10
1
10
6 
x 10
5
x
4
2
3x
4 2
81x 2
1
3x
3
5
3
4 
x
2
1
x
3 2
12
x
x  12
8
12
1 
x 12
9
x 4
3
x
3
2
12
1
x
x

13. 40 
Examples:
4 10
 
If and are real numbers, then a b
a b a b
  
Product Rule for Square Roots
2 10
7 75  7 25 3
  7 5 3
  35 3
Simplifying Rational Expressions

17
16x 
x
x16
16 x
x8
4

3 17
16x 

3 2
15
2
8 x
x 3 2
5
2
2 x
x

10
4

3
25
7

14. 16
81

Examples:
2
5
4
9
45
49

a
If and are real numbers and 0,then
b
a
a b b
b
 
Quotient Rule for Square Roots
2
25

9 5
7


3 5
7
16
81

2
25

45
49

Simplifying Rational Expressions

15. 15
3

90
2

a
If and are real numbers and 0,then
b
a
a b b
b
 
3 5
3


3 5
3

 5
9 10
2


9 2 5
2
 

9 2 5
2
 
 3 5
Simplifying Rational Expressions

16. 11
x 
Examples:
7
7
25
y

8
27
x

6
7
25
y y


3
7
5
y y
10
x x
 
5
x x
4
18x  4
9 2x
  2
3 2
x
8
9 3
x

 4
3 3
x
8
27
x

Simplifying Rational Expressions

17. 3
88 
Examples:
3
81
8

3
10
27

3
3
81
8

3
27 3
2


3
8 11
  3
2 11
3
10
3
3
3
10
27

3
3 3
2
Simplifying Rational Expressions
3 3 7
27m n  3 3 6
3 m n n  2 3
3mn n

18. One Big Final Example
12 4 18
5
64x y z 
10 2 4 15 3
5
32 2x x y z z
 
2 3 2 4 3
5
2 2
x z x y z
Simplifying Rational Expressions

19. 5 3
x x
 
Review and Examples:
6 11 9 11
 
8x
15 11
12 7
y y
  5y
7 3 7
  2 7

Adding, Subtracting, Multiplying Radical
Expressions

20. 27 75
 
Simplifying Radicals Prior to Adding or Subtracting
3 20 7 45
 
9 3 25 3
   
3 4 5 7 9 5
   
3 3 5 3
  8 3
3 2 5 7 3 5
   
6 5 21 5
  15 5

36 48 4 3 9
    6 16 3 4 3 3
    
6 4 3 4 3 3
    3 8 3

Adding, Subtracting, Multiplying Radical
Expressions

21. 4 3 3
9 36
x x x
  
Simplifying Radicals Prior to Adding or Subtracting
6 6
3 3
10 81 24
p p
 
2 2 2
3 6
x x x x x
  
2
3 6
x x x x x
  
2
3 5
x x x

6 6
3 3
10 27 3 8 3
p p
   
2 2
3 3
10 3 3 2 3
p p
  
2 3
28 3
p
2 2
3 3
30 3 2 3
p p
 
Adding, Subtracting, Multiplying Radical
Expressions

22. 5 2
 
7 7
 
10 2
x x
 
If and are real numbers, then a b
a b a b
  
10
49  7
6 3
  18  9 2
  3 2
2
20x  2
4 5x
  2 5
x
Adding, Subtracting, Multiplying Radical
Expressions

23.  
7 7 3
  7 7 7 3
    49 21
 
 
5 3 5
x x  
  
5 3
x x
  
7 21

2
5 3 25
x x
  5 3 5
x x
  
5 15
x x

2
3 5 15
x x x
   
2
3 5 15
x x x
  
Adding, Subtracting, Multiplying Radical
Expressions

24.   
3 6 3 6
  
 
2
5 4
x  
9 6 3 6 3 36
    3 36
 
33

  
5 4 5 4
x x
  
2
25 4 5 4 5 16
x x x
   
5 8 5 16
x x
 
Adding, Subtracting, Multiplying Radical
Expressions

25. 1
6

Dividing Radical Expressions
4
3 2

