Section 10.3 – 10.4

Multiplying and Dividing Radical Expressions

QuestionsQ: True or False?

Product /Quotient Rule for Radicals

9494 True False

9494 True False

9494 True False

9

4

9

4 True False

Product and Quotient Rules

1) The power of each factor in the radical is less than the index

2) The radicand contains no fractions or negative numbers

3) No radical appears in the denominator.

nnn baab

where a, b are non-negative numbers

n

n

n

b

a

b

a

A radical expression is in simplified form if

ExamplesSimplify the following expressions

98

3 5m

3 754y

236

11

a

4 158ut

a

ab

2

50 2

3 314189 dc

Solution

Divide and, if possible, simplify.

4 53

3

48a)

3

24b)

2 3

x y

x

48 48a)

33

16 4

Because the indices match, we can divide the radicands.

Example

Solution continued

4 5 4 533

324 1 24

b) 2 32 3

x y x y

xx

3 5318

2x y

3 3 23 318

2x y y

2312

2xy y

23xy y

Rationalizing Denominators or Numerators With One Term

When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator no longer contains a radical. The procedure for finding such an expression is called rationalizing the denominator.

Solution

Rationalize each denominator.

23

5a)

33

b) 4

xy

xy

5 5a)

3 3x x 35

3 3

x

x x

215

3

x

x 15

3

x

x

Multiplying by 1

Example

Solution

3

2

22 23

3

3

3 3b

2

2)

4 4

y y

xy x

x y

x yy

23

3 33

3 2

8

y x y

x y

2 23 33 2 3 2

2 2

y x y x y

xy x

Property of radicals

when n is odd

3 34

2

4

6

5 5)2(

4 42

7 76

2)5(

6 6)2(

2

5

2

n nx x

when n is even

n nx x

Evaluate the radical expressions

5 5a

4 43)(y

a

3y4 12y

4

42

w

3 3)( vu

44

16

w

62436 cba

vu

w

2

232 )6( bca326 bca