SIMPLIFY RADICAL EXPRESSIONS

Simplify radical expressions :

Before learning how to simplify radical expressions, we must know about like radicals and unlike radicals. 

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. 

Let us see some example problems to understand this topic

Example 1 :

The value of √20 - √225 + √80

Solution :

√20  =  √2 ⋅ 2 ⋅ 5  =  2√5

√225  =  √5 ⋅ 5 ⋅ 3 ⋅ 3  =  5 ⋅ 3  =  15

√225  =  √2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5  =  (2 ⋅ 2)√5  =  4√5

Then, we have

 √20 - √225 + √80  =  2√5 - 15 + 4 √5

√20 - √225 + √80  =  6√5 - 15

   √20 - √225 + √80  =  6 √5  -  15

In (6 √5  -  15), both 6 and 15 are multiples of 3. 

So, we can factor out 3 from 6 and 15. 

√20 - √225 + √80  =  3(√5  -  5)

Hence the answer is 3(2√5 - 5).

Example 2 :

Simplify the following radical expression

√27 + √75 + √108 - √48

Solution :

First we have to decompose each term in the radical expression above as much as possible.

So, we have

=  √27 + √75 + √108 - √48

=  √(3 ⋅ 3 ⋅ 3) + √(3 ⋅ 5 ⋅ 5) + √(3 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 2) - √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)

Simplify.

=  3√3  +  5√3  +  2 ⋅ 3 ⋅ √2  -  2 ⋅ 2 ⋅ √3

=  3√3  +  5√3  +  6√2 - 4√3

=  3√3  +  5√3  -  4√3  +  6√2.

=  4√3 + 6√2 

In (4√3 + 6√2), both 4 and 6 are multiples of 2.

So, we can factor out 2. Then, we get

=  2(2√3 + 3√2) 

Hence  2(2√3 + 3√2) is the answer.

Example 3 :

Simplify the following radical expression

5√28 - √28 + 8 √28

Solution :

=  5√28 - √28 + 8 √28

Because all the terms in the above radical expression are like terms, we can simplify as given below. 

=  12√28

=  12√(2 ⋅ 2 ⋅ 7)

=  12 ⋅ 2√7

=  24√7

Hence  24 √7  is the answer.

Example 4 :

Simplify the following radical expression

9√11 - 6√11 

Solution :

=  9√11 - 6√11 

Because the terms in the above radical expression are like terms, we can simplify as given below. 

=  3√11

Hence  3√11 is the answer.

Example 5 :

Simplify the following radical expression

7√8 - 6√12 + 5√32

Solution :

=  7√8 - 6√12 - 5√32

Since they are not like radicals we have to split the numbers inside the radicals as much as possible to make them as like radicals.

=  7 √(2 x 2 x 2) - 6 √(2 x 2 x 3) - 5 √(2 x 2 x 2 x 2 x 2)

=  (7 x 2) √2 - (6 x 2) √3 - (5 x 2 x 2) √2

=  14 √2 - 12 √3 - 20 √2

=  (14 - 20) √2 - 12 √3

=  -6 √2 - 12 √3

Hence  -6 √2 - 12 √3 is the answer.

Example 6 :

Simplify the following radical expression

2 √99 + 2 √27 - 4 √176 - 3√12

Solution :

=  2 √99 + 2 √27 - 4 √176 - 3√12

Since they are not like radicals we have to split the numbers inside the radicals as much as possible to make them as like radicals.

=  2 √(3 x 3 x 11) + 2 √(3 x 3 x 3) - 4 √(2 x 2 x 2 x 2 x 11) -

3 √(2 x 2 x 3)

=  (2 x 3) √11 + (2 x 3) √3 - (4 x 2 x 2) √11 - (3 x 2) √3

=  6 √11 + 6 √3 - 16 √11 - 6 √3

=  6 √11  - 16 √11 + 6 √3 - 6 √3

=  (6 - 16) √11  + (6 - 6) √3

=  -10 √11  + 0 √3

=  -10 √11

Hence -10 √11 is the answer.

Related pages

• Properties of radicals
• Simplifying radical expressions worksheets
• Square roots
• Ordering square roots from least to greatest
• Operations with radicals
• How to simplify radical expressions

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