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