Simplifying radicals with variables and exponents worksheet

Key Concept

If the given radical is square root, write each term inside the radical as squares. 

If the given radical is cube root root, write each term inside the radical as cubes. 

If it is square root, we can get one term out of the radical for every two same terms multiplied inside the radical. 

If it is cube root, we can get one term out of the radical for every three same terms multiplied inside the radical. 

Four fourth root and more, we have to do as explained for square root and cube root above. 

Example 1 :

Simplify :

√(16u4v3)

Solution : 

=  √(16u4v3)

=  √(42 ⋅ u2 ⋅ u2 ⋅ v2 ⋅ v)

=  (4 ⋅ u ⋅ u ⋅ v)√v

=  4u2v√v

Example 2 :

Simplify :

√(147m3n3)

Solution : 

=  √(147m3n3)

=  √(3 ⋅ 72 ⋅ m2 ⋅ m ⋅ n2 ⋅ n)

=  (7 ⋅ m ⋅ n)√(3mn)

=  7mn√(3mn)

Example 3 :

Simplify :

√(75x2y)

Solution : 

=  √(75x2y)

=  √(3 ⋅ 52 ⋅ x2 ⋅ y)

=  (5 ⋅ x)√(3y)

=  5x√(3y)

Example 4 :

Simplify :

6√(72x2)

Solution : 

=  6√(72x2)

=  6√(2 ⋅ 62 ⋅ x2)

=  (6 ⋅ 6 ⋅ x)√2

=  36x√2

Example 5 :

Simplify :

3√(8x6y3)

Solution : 

=  3√(8x6y3)

=  3√(23 ⋅ x3 ⋅ x3 ⋅ y3)

=  2 ⋅ x ⋅ x ⋅ y

=  2x2y

Example 6 :

Simplify :

3√(54m5n6)

Solution : 

=  3√(54m5n9)

=  3√(2 ⋅ 33 ⋅ m3 ⋅ m2 ⋅ n3 ⋅ n3 ⋅ n3)

=  (3 ⋅ m ⋅ n ⋅ n ⋅ n)3√(2 ⋅ m2)

=  3mn3√(2m2)

Example 7 :

Simplify :

4√(81m4n8)

Solution : 

=  4√(81m4n8)

=  4√(34 ⋅ m4 ⋅ n4 ⋅ n4)

=  3 ⋅ m ⋅ n ⋅ n

=  3mn2

Example 8 :

Simplify :

5√(32p10q15)

Solution : 

=  5√(32p10q15)

=  5√(25 ⋅ p5 ⋅ p5 ⋅ q5 ⋅ q5 ⋅ q5)

=  2 ⋅ p ⋅ p ⋅ q ⋅ q ⋅ q

=  2p2q3

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to 

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

How do you do radical and rational exponents?

How do you solve a radical equation?