SIMPLIFYING RADICAL EXPRESSIONS WITH VARIABLES

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

Like Radicals :

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

Unlike Radicals :

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

We can add and subtract like radicals only.

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

Step 1 :

Decompose the number inside the radical into prime factors. 

Step 2 :

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. 

Step 3 :

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.

Step 4 :

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.

Step 5 :

Combine the radical terms using mathematical operations. 

Example : 

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

 √18 + √8  =  3√2 + 2√2

 √18 + √8  =  5√2

Solved Questions

Question 1 :

Simplify : 

√(16u⁴v³)

Solution :

√(16u⁴v³)  =  √(4 ⋅ 4 ⋅ u² ⋅ u² ⋅ v ⋅ v ⋅ v)

√(16u⁴v³)  =  4u²v√v

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) / ⁴√16

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)  =  12y√2

Question 6 :

Find the square root of (196a⁶b⁸c¹⁰).

Solution :

√(196a⁶b⁸c¹⁰)  =  √(14 ⋅ 14 ⋅ a³ ⋅ a³ ⋅ b⁴ ⋅ b⁴ ⋅ c⁵ ⋅ c⁵)

√(196a⁶b⁸c¹⁰)  =  14a³b⁴c⁵

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.