Multiplying, Dividing, and Simplifying Radicals10.310.3

1. Multiply radical expressions.2. Divide radical expressions.3. Use the product rule to simplify radical expressions.

254 1052 100 10

Product Rule for Radicals

If both and are real numbers, then

.n n na b a b

n a n b

The index (root) must be the same!

123 636

Multiply:Multiply:

ba 53 ab15

75 35

33 7xxy 3 27 yx

2516

54

2516

54

The index (root) must be the same!

, where 0.n

nn

a ab

bb

Quotient Rule for RadicalsIf both and are real numbers, thenn a n b

6449

87

Divide:Divide:

8115

915

Divide:Divide:

3243

3243 81 9

321 7

321

149162536496481100

Simplify radicals:Simplify radicals:

20 54 52

Look for the largest perfect square that divides into the radicand evenly.

54

75 325 35325

60 154 152154

48 316 34316

Simplify radicals:Simplify radicals:

245

149162536496481100

Create a factor tower.

3339

20

10202

522

52

20

5102202

20 52

Divide by prime numbers!

2, 3, 5, 7, 11, …

60

30602

5322

15302602

5153302602

532

152

60

60 152

48

24482

32222

12242482

6122242482

322

34

48

362

122242482

48 34

Simplify radicals:Simplify radicals:

245

149162536496481100

245

7497

245549

2455

775

57

149162536496481100

Simplify radicals:Simplify radicals:

300 3103100

5255753

15023002

310

352

149162536496481100

Simplify radicals:Simplify radicals:

52 132134

13262522 132

149162536496481100

Simplify radicals:Simplify radicals:

108 36336

393

273542

1082

36

332

Slide 10- 16Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

486

6 9

3 54

9 6

18 27

Slide 10- 17Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

486

6 9

3 54

9 6

18 27

Simplify radicals:Simplify radicals:

3x xxxxx

5x xxx xxxxx

xx2

1132R

Outsid

e

Insid

e

1252R

Outsid

e

Insid

e

Think…

Simplify radicals:Simplify radicals:

9x

1492R

Outsid

e

Insid

eThink:

xx4

Simplify:Simplify:

3775 yx 3775 yx

25∙3

35 xx 35 3 xyyx 35 3

1372R 11

32R

Simplify:Simplify:

47348 cba 47348 cba

16∙3

34 aa 34 abab 34 3

0242R11

32R 13

72R

abcab 34 23

3 25

5255

12552502

182764125216343

Simplify radicals:Simplify radicals:

3 250 3 2125

3 25

393

273813

2433

182764125216343

Simplify radicals:Simplify radicals:

3 243 3 933 927

3 93

Simplify:Simplify:

3 52724 zyx 3 53 23 73 24 zyx

3

8∙3

3 32 xx 32 2 3 22 32 xyx

2153R12

73R 20

23R

3 222 32 zxyzx

Simplify:Simplify:

52 102156 cc

715012 c

1225∙6

cc 6512 3 6512

149162536496481100

cc 660 3

Simplify:Simplify:46

109

59

24054

sr

sr

46

109

5240

954

srsr

63486 sr16∙3

rr 346 346

149162536496481100

rrs 324 3

6 rrs 346 3

Slide 10- 27Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

2 33 9m m

3 3m m

23 3m m

627m

2 33 3m m

Slide 10- 28Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

2 33 9m m

3 3m m

23 3m m

627m

2 33 3m m