Section 7.4 Multiply & Divide Radical Expressions

Multiplying Radical Expressions

Powers of Radical Expressions

Rationalizing Denominators

Rationalizing 2-Term Denominators

Rationalizing Numerators

Multiplying a Monomial by a Monomial

Use the commutative and associative properties of

multiplication to multiply the coefficients and the

radicals separately. Then simplify any radicals in

the product, if possible.

Product Rule:

141027)5)(2()25)(72(

nnn baba

Multiplying a Polynomial by a Monomial

To multiply a polynomial by a monomial, use the

distributive property to remove parentheses and

then simplify each resulting term, if possible.

3210121681012)8253)(24(

Using FOIL to Multiply Radicals

To multiply a binomial by a binomial,

we use the FOIL method.

32

33

)3)(1(

xx

xxxx

xx

Examples for You

Rationalizing a Denominator

(leave no radical in the denominator)

A radical expression is in simplified form when

each of the following statements is true.

1. No radicals appear in the denominator of a

fraction.

2. The radicand contains no fractions or negative

numbers.

3. Each factor in the radicand appears to a power

that is less than the index of the radical.

Dividing and Rationalizing the

Denominator

To divide radical expressions, rationalize the denominator

of a fraction to replace the denominator with a rational

number.

To eliminate the radical in the denominator, multiply the

numerator and the denominator by a number that will give

a perfect square under the radical in the denominator.

5

302

5

532

)5(5

)5(24

5

242

3

3

275

3

3

3

5

3

5 4

4 3

4 3

44

Simplifying Before Rationalizing

22

3 2

3 66

3 2

3

22

22

44

3

44

3

4 2

4

2

4

2

2

2

1

2

1

16

1

h

h

h

h

h

h

hhh

Practice

Rationalizing Monomial Denominators

393

27

3

27

3

36

6

62

6

6

6

2

6

233 2

3 2

3 2

33

y

y

xy

yx

xy

yx

yx

yx

xy

x

xy

x 3 23 23 234

3 222

3 222

3

3

3

33

3

33

3

3

3

3

3

9

3

9

Rationalizing Binomial Denominators

To rationalize a denominator that contains a binomial

expression involving square roots, multiply its numerator

and denominator by the conjugate of its denominator.

Conjugate binomials are binomials with the same terms but

with opposite signs between their terms

1313

)13(2

)13(

)13(

)13(

2

13

2

Practice

Rationalizing Binomial Denominators

6323

)23(3

23

23

23

3

23

3

yx

yyx

yx

yx

yx

yx

yx

yx

2

Rationalizing Numerators

In calculus, we sometimes have to rationalize a numerator by multiplying the numerator and denominator of the fraction by the conjugate of the numerator.

xx

x

x

x

x

x

x

x

3

9

)3(

)3(33

Section 9.5 Multiplication and

Division of Radical Expressions

Example 1

7253 356

Example 2

125623

365182

65292

30232 3026

Example 3

53453

94 15 154 25

)5(153)3(4

515312

1537

Example 4 Expand

33 xx

2x x3 x3 9

96 xx

23x

Example 5 Expand

yxyx 2323

29 x xy6 xy6 24 y

yxyx 4129

223 yx

Example 6 Expand

1212 xx

2)2( x 2 x 1

1222 xx

212 x

2 x

223 xx

Example 7 Multiply

2626

26 122212

6 2

4

Notice how the radicals do not

appear in the final answer.

This is important for the next

problem.

Example 8 Rationalize the denominator

35

6

35

35 This conjugate factor is

another form of the

number 1

22 315155

356

35

356

2

356

1

353 3353

Multiplication and

Division of Radicals

Objectives:

Recall how to simplify radicals, add and

subtract radicals

Multiply radicals with same index

Divide radicals with same index, rationalizing

the denominator

Multiply radicals with different index

Simplify, then add or subtract:

40510006

25816

185243

16546

20556 516

1070

21566

32

227

Review: Simplifying radicals Quantitative Relationship

Given two quantities Q1 and Q2, determine the relationship of Q1 to Q2.

Use the following options:

A: If Q1 > Q2

B: If Q1 < Q2

C: If Q1 = Q2

D: Relationship cannot be determined

1253 27

4 123 125

3 84 16

5 32 3 12a

45 32

Quantity 1 Quantity 2

A

A

B

D

C

*To multiply radicals: multiply the

coefficients (if any), multiply the

radicands of the same index, and then

simplify the product if possible.

k kX Y

35*5 175 7*25 75

Let’s Try!

73*82 566 14*46

142*6 1412

204*52 1008 8010*8

2

5 5*5 25 5

2

7 7*7 49 7

2

8 8*8 64 8

2

x xx * 2x x

To divide radicals: Divide

the coefficients (if any),

divide the radicands of the

same index (if possible),

and rationalize the

denominator so that no

radical remains in the

denominator

k

k

X

Y

7

568 2*4 22

7

6

This cannot be divided

which leaves the radical

in the denominator. We

do not leave radicals in

the denominator. So we

need to rationalize by

multiplying the fraction

by something so we can

eliminate the radical in

the denominator.

7

7*

7

6

49

42

7

42

42 cannot be simplified,

so we are done.

This can be divided

which leaves the

radical in the

denominator. We do

not leave radicals in

the denominator. So

we need to

rationalize by

multiplying the

fraction by

something so we can

eliminate the radical

in the denominator.

10

5

2

2*

2

1

2

2

This cannot be

divided which

leaves the radical in

the denominator.

We do not leave

radicals in the

denominator. So we

need to rationalize

by multiplying the

fraction by

something so we

can eliminate the

radical in the

denominator.

12

3

3

3*

12

3

36

33

6

33

2

3

Reduce

the

fraction.

Multiplication of Radicals

with Different Indices

How do we multiply radicals

with different indices?

Multiply

1. )2)(3( 3)2)(3( 3

1

2

1

)2)(3( 6

2

6

3

)2)(3( 6 26 3

6 23 )2)(3(

6 )4)(27(

6 108

.2 )(23 yx

))(2())(2( 33 yx

))(2())(2( 2

1

3

1

2

1

3

1

yx

))(2())(2( 6

3

6

2

6

3

6

2

yx

)3)(2())(2( 66 26 36 2 yx

6 36 3 44 yx

We apply distributive property

Seatwork: On bondpaper

Copy and answer.

)126()153( xx 1.

2. )10(303 3 yxy

3. 3 32

3 65

4

32

yx

yx

4. 3 25 4 xx

5. 3 52

5108 x

230303 yyxy

xy2

15 715 22 xxx

66 23 25852

ANSWERS:

6 200

More on Rationalizing the denominator. Example a.

5 28

3

x 5 232

3

x

5 3

5 3

4

4

x

x

Think about

this. 5523 2?)2( xx

332 42 : xxAnswer

x

x

2

4 3 5 3

Example b.

25

3

Try this:

)25)(25(

Answer:

4101025

425

25& 25 are called conjugates. Therefore, to

rationalize the denominator (in this

case), we multiply by its conjugate.

25

25

2

)2353

2

)25(3

Using FOIL method

253

Let’s try this out!

3 5

3

16

3 )1

x 532

2

3 54

3

2

3

x

3 2

3

22

3

xx

3

3

4

4

x

x

)2)(2(

123

xx

x

2

3

4

12

x

x

532

532

2594

1062

512

1062

7

1062

))(( 532 152

))(( 6268 96

))(( 7473 84

))(( 15354 360

)()( 6472 428

Perform the indicated operation and simplify:

75556 531

3 1615331 3 230331

)75)(56( 1530

2)33( 27

2)253( 51249

What Next?

Section 7.5

Radical Equations