Math Skill: Multiplying and Dividing Radical Expressions
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Knowing how to simplify radical expressions will make multiplying and dividing radical expression easier.
To review radical expression simplification, see here.

From simplification, we know that √ab = √a • √b

Examples

Solve and simplify.
a.
 √3 • √7 = √3 • 7 Multiply = √21
b.
 √5 • √15 = √5 • 15 Multiply = √5 • 5 • 3 Factor to find perfect square factors = 5√3
c.
 √6x2y • √27x6 = √6x2y • 27x6 Multiply = √6 • 27 • √x2 • x6 • √ y Separate integers and variables to make simplifying easier = √2 • 3 • 3 • 9 • √x8 • √ y Factor to find perfect square factors = 9√2 • x4 • √ y Simplify = 9x4√2y Combine

For any nonnegative numbers a and b, where b ≠ 0,
=

Examples

Solve and simplify.
a.
= = 25 = 5
b.
= = 6b

Rationalizing the denominator

Removing a radical from the denominator is called rationalizing the denominator.

Examples

Solve and simplify.
a.
=
 •
Multiply by
 √2 √2
= 1
=
b.
=
 •
Multiply by
 √2 √2
= 1
=
c.
=
 •
Multiply by
 √2y √2y
= 1
=
=

Complexity=1, Mode=mul

Solve and simplify.
1.
 √3 • √15 =
2.
 √7 • √35 =

Complexity=2, Mode=mul

Solve and simplify.
1.
 9e 2
 9e 10
=
2.
 5a 9
 4a 2
=

Complexity=1, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.

=
2.
=

Complexity=2, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.
=
2.
=

Complexity=3, Mode=mul

Solve and simplify.
1.
21r
 14r 8s 7
=
2.
 10mn 9
 30m 3n
=

Complexity=3, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.

=
2.
=

Complexity=1, Mode=mul

Solve and simplify.
1
 √3 • √15 =
Solution
 3
 15
=
 3 • 15
=
 3 • 3 • 5
=3
 5
2
 √7 • √35 =
Solution
 7
 35
=
 7 • 35
=
 7 • 7 • 5
=7
 5

Complexity=2, Mode=mul

Solve and simplify.
1
 9e 2
 9e 10
=
Solution
 9e 2
 9e 10
=
 9 • e 2
 9 • e 10
=
 9 • 9
 e 2 • e 10
=
9
 e 12
=
 9 • e 6
=9e 6
2
 5a 9
 4a 2
=
Solution
 5a 9
 4a 2
=
 5 • a 9
 4 • a 2
=
 4 • 5
 a 9 • a 2
=
2
 5
 a 11
=
2
 5
a 5
 a
=2a 5
 5a

Complexity=1, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1

=
Solution

=
=
2
=
Solution
 = =

Complexity=2, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1
=
Solution
=
=
 •
=
2
=
Solution
The numerator and denominator are both divisible by 2
=

=

=
=
 •
=

Complexity=3, Mode=mul

Solve and simplify.
1
21r
 14r 8s 7
=
Solution
 21r
 14r 8s 7
=
 7 • 3 • r
 7 • 2 • r 8 • s 7
=
 2 • 3 • 7 • 7
 r • r 8
 s 7
=
7
 2 • 3
 r 9
s 3
 s
=
7
 6
r 4
 r
s 3
 s
=7r 4s 3
 6rs
2
 10mn 9
 30m 3n
=
Solution
 10mn 9
 30m 3n
=
 10 • m • n 9
 10 • 3 • m 3 • n
=
 3 • 10 • 10
 m • m 3
 n 9 • n
=
10
 3
 m 4
 n 10
=
10
 3
m 2n 5
=10m 2n 5
 3

Complexity=3, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1

=
Solution
The numerator and denominator are both divisible by 2s

=
 2s 9
=s 42s