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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