
Simplify
 2 x y 
3x y 
3
2
4
5 3
Section P.3

How do we simplify expressions involving
radicals and/or rational exponents?

If a≥0 and b≥0 and b2 = a, then b is the
principal square root of a.
b a  b a
2
2
a a

If a≥0 and b≥0, then
ab  a b and
a b  ab
• The square root of a product is the product
of the square roots.

Simplify
a) 500
b) 6 x 3 x
Solution:
a.
500  100 5
b.
6x  3x  6x  3x
 100 5
 18x 2  9x 2 2
 10 5
 9x 2 2  9 x 2 2
 3x 2

Simplify
a) 75
b) 5 x 15 x

If a≥0 and b>0, then
a
a

b
b
and
a

b
a
.
b
• The square root of the quotient is the
quotient of the square roots.

Simplify:
100
9
Solution:
100
100 10


9
9
3

Simplify: a)
49
16
b)
36
144

Read Section P.3
Page 32 #1-25 odd

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
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
Evaluate each expression in Exercises 1-6 or
indicate that the root is not a real number.
1)

36
3)
 36
13
2
5)
Use the product rule to simplify the
expressions in Exercises 7-16. In Exercises
11-16, assume the variables represent
nonnegative real numbers. 7)
50
9)
11)
45 x 2
2x  6x
3
13)
x
15)
2x2  6x

Simplify
4 3 32 3
3 3

Simplify
2 5 5 5  5
4 5

Simplify
24  2 6 
4 6

Simplify
18  5 8 
13 2
a)
12 x 15 x
25 36
b) 3 54  2 24  96  4 63  2 28



We DO NOT leave radicals in the denominator
Multiply numerator and denominator by the
smallest number that will eliminate the
radical.
If square root: can multiply top and bottom
by the radical in the denominator, then
simplify
a)
6
18
b)
7
3
a)
5
3
b)
6
12



For a denominator of form a  b , we
multiply numerator and denominator by its
conjugate, a  b


5
3 7
4
6 5
n

a b
n
means that b  a
If n, the index, is even, then a > 0 and b > 0.
If n is odd, a and b can be any real numbers.

For all real numbers, where the indicated
roots represent real numbers,
n
a b  ab and
n
n
n
n
a n a

, b0
b
b
3
a) 54
b) 8  8
4
4
8
c) 3
125
3
a) 40
b) 8  8
5
5
27
c) 3
1000
3 32  2 2
4
4
3 81  4 3
3
3



Read Section P.3
Page 32 #27-75 odd
You have until 1:50 to work on this
assignment. We will then finish the P.3 notes.
a
1 /n
 n a.
Furthermore,
1
1
1/ n
a
 1/ n  n , a  0
a
a
a) 4
1
2
b) 16
1

4
c) 27
1
3
1
4
a) 81
1

3
b) 125
c) 64
1
3
a

m/ n
m
n
m
( a)  a .
n
The exponent m/n consists of two parts: the
denominator n is the index of the radical
and the numerator m is the exponent.
Furthermore,
a
m/n

1
a
m/n
.
a) 9
3
2
b) 125
2
3
c) 16
3

4
a) 4
3
2
b) 32
2

5
c) 8
5

3

a) 3x
34
2 x 
12
43
12 x
b) 1 4
6x

a) 2 x
43
5x 
83
4
20 x
b) 3 2
5x

Page 32 #77-93 odd, 104, 106, 121

In Exercises 77-84, evaluate each expression
without using a calculator.
77) 361 2

79) 81 3
83) 324 5
81) 1252 3
In Exercises 85-94, simplify using properties
of exponents.
3
85)
7 x 2 x 
13
12
87)
20 x
14
5x
14
89)
91)
x 
25 x y 
3 y 
23
4
6 12
14 3
93)
y
1 12