# SIMPLIFYING RADICAL EXPRESSIONS WITH VARIABLES

Simplifying Radical Expressions with Variables :

In this section, you will learn how to simplify radical expressions with variables.

The radicals which are having same number inside the root and same index is called like radicals.

Unlike radicals don't have same number inside the radical sign or index may not be same.

## Simplifying Radical Expressions - Steps

Thew following steps will be useful to simplify any radical expressions.

(i) Decompose the number inside the radical into prime factors.

(ii) If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical.

(iii) If you have cube root (3), you have to take one term out of cube root for every three same terms multiplied inside the radical.

(iv) If you have fourth root (4), you have to take one term out of fourth root for every four same terms multiplied inside the radical.

(v) Combine the radical terms using mathematical operations.

Example :

√18 + √8  =  √(3 ⋅ 3 ⋅ 2) + (2 ⋅ 2 ⋅ 2)

√18 + √8  =  32 + 22

√18 + √8  =  52

## Simplifying Radical Expressions with Variables - Practice Questions

Question 1 :

Simplify :

√(16u4v3)

Solution :

√(16u4v3)  =  √(4 ⋅ 4 ⋅ u⋅ u⋅ v ⋅ v ⋅ v)

√(16u4v3)  =  4u2vv

Question 2 :

Simplify :

√(147m3n3)

Solution :

√(147m3n3)  =  √(7 ⋅ 7 ⋅ 3  m ⋅ m ⋅ m ⋅ n ⋅ n ⋅ n)

√(147m3n3)  =  7mn√(3mn)

Question 3 :

Simplify :

3√(125p6q3)

Solution :

3√(125p6q3)  =  3√(5 ⋅ 5 ⋅ 5  p2 ⋅ p2 ⋅ p2 ⋅ q ⋅ q ⋅ q)

3√(125p6q3)  =  5p2q

Question 4 :

Simplify :

4√(x4/16)

Solution :

4√(x4/16)  =  4√(x4) / 416

4√(x4/16)  =  4√(x ⋅ x ⋅ x ⋅ x) / 4√(2 ⋅ 2 ⋅ 2 ⋅ 2)

4√(x4/16)  =  x / 2

Question 5 :

Simplify :

6√(72y2)

Solution :

6√(72y2)  =  6√(6 ⋅ 6 ⋅ 2  y ⋅ y)

6√(72y2)  =  6(6y)2

6√(72y2)  =  12y2

Question 6 :

Find the square root of (196a6b8c10).

Solution :

√(196a6b8c10)  =  √(14 ⋅ 14 ⋅ a a3 ⋅ b⋅ b⋅ c⋅ c5)

√(196a6b8c10)  =  14a3b4c5

After having gone through the stuff above, we hope that the students would have understood how to simplify radical expressions with variables.

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