Name
Class
6-2
Date
Notes
Multiplying and Dividing Radical Expressions
You can simplify a radical if the radicand has a factor that is a perfect nth power and n
is the index of the radical. For example:
n xy n z  y n xz
What is the simplest form of each product?
a. 3 12  3 10
3 12  3 10  3 12  10
Use n a  n b  n ab .
3
 22  3  2  5
3
 23  3  5
3
 23  3 3  5
Write as a product of factors.
Find perfect third powers.
Use n ab  n a  n b .
 23 15
b.
Use
n a n  a to simplify.
7 xy3  21xy 2
7 xy3  21xy 2  7 xy3  21xy 2
Use n a  n b  n ab .
 7 xy 2 y  3  7 xy 2
Write as a product of factors.
 72 x 2 ( y 2 )2  3 y
Find perfect second powers.
 7 xy 2 3 y
Use
n a n  a to simplify.
Exercises
Simplify each product.
1.
15x  35x
4. 5 7 x3 y  28 y 2
2. 3 50 y 2  3 20 y
3. 3 36 x2 y5  3 6 x2 y
5. 3 9 x5 y 2  3 2 x2 y5
6.
3( 12  21)
Name
Class
Date
Notes (continued)
6-2
Multiplying and Dividing Radical Expressions
Rationalizing the denominator means that you are rewriting the expression so
that no radicals appear in the denominator and there are no fractions inside
the radical.
9y
What is the simplest form of
2x
?
Rationalize the denominator and simplify. Assume that all variables are positive.
9y
9y

Rewrite as a square root of a fraction.
2x
2x

9 y  2x
Make the denominator a perfect square.
2x  2x

18 xy
4 x2
Simplify.

18 xy
22  x 2
Write the denominator as a product of perfect squares.

18 xy
2x
Simplify the denominator.
32  2  x  y
2x
3 2 xy

2x

Simplify the numerator.
Use
n a n  a to simplify.
Exercises
Rationalize the denominator of each expression. Assume that all variables are
positive.
7.
11.
5
x
3
4 k9
3
16 k 5
3
8.
12.
6ab2
3 4
2a b
3x5
5y
9.
13.
4 9y
4x
4 10
4 2
z
10.
10 xy3
12 y 2
19a 2b
14. 3
abc 4