SIMPLIFY RADICAL EXPRESSIONS

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

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.

The following steps will be useful to simply radical expressions

Step 1 :

Decompose the number inside the radical sign into prime factors.  

Step 2 :

Take one number out of the radical for every two same numbers multiplied inside the radical sign, if the radical is a square root.

Take one number out of the radical for every three same numbers multiplied inside the radical sign, if the radical is a cube root.

Step 3 :

Simplify.

Examples :

√4  =  √(2  2)  =  2

√16  =  √(2  2  2  2)  =  2  2  =  2

3√27  =  3√(3  3  3)  =  3

3√125  =  3√(5  5  5)  =  5

Question 1 :

Simplify :

20 - 225 + 80

Solution :

Decompose 20, 225 and 80 into prime factors using synthetic division. 

√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

20 - 225 + 80  =  3(2√5 - 5)

Question 2 :

Simplify :

√27 + √75 + √108 - √48

Solution :

Decompose 27, 75, 48 and 108 into prime factors using synthetic division. 

√27  =   √(3  3  3)  =  3√3

√75  =   √(5  5 ⋅ 3)  =  5√3

√108  =   √(3  3  3 ⋅ 2 ⋅ 2)  =  3 ⋅ 2 ⋅ √3  =  6√3

√48  =  √(2  2  2  2  3)  =  2 ⋅ 2 ⋅ √3  =  4√3

Then, we have

√27 + √75 + √108 - √48  =   3√3 + 5√3 + 6√3 - 4√3

√27 + √75 + √108 - √48  =   10√3

Question 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. 

5√28 - √28 + 8√28  =  12√28

5√28 - √28 + 8√28  =  12√(2 ⋅  7)

5√28 - √28 + 8√28  =  12 ⋅ 2√7

5√28 - √28 + 8√28  =  24√7

Question 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. 

9√11 - 6√11  =  3√11

Question 5 :

Simplify the following radical expression

7√8 - 6√12 - 5√32

Solution :

7√8 - 6√12 - 5√32

Decompose 8, 12 and 32 into prime factors.

7√8  =  7√(2  2  2)  =  7 ⋅ 2√2  =  14√2

6√12  =  6√(2  2 ⋅ 3)  =  6 ⋅ 2√3  =  12√3

5√32  =  √(2  2  2 ⋅ 2 ⋅ 2)  =  5 ⋅ ⋅ 2 ⋅ √2  =  20√2

Then, we have

7√8 - 6√12 + 5√32  =  14√2 - 12√3 - 20√2

7√8 - 6√12 + 5√32  =  14√2 - 12√3 - 20√2

7√8 - 6√12 + 5√32  =  -6√2 - 12√3

7√8 - 6√12 + 5√32  =  -6(√2 + 2√3)

Question 6 :

Simplify the following radical expression

2√99 + 2√27 - 4√176 - 3√12

Solution :

Decompose 99, 27, 176 and 12 into prime factors.

2√99  =  2√(3  3  11)  =  2 ⋅ 3√11  =  6√11

2√27  =  2√(3  3 ⋅ 3)  =  2 ⋅ 3√3  =  6√3

4√176  =  √(2  2  2 ⋅ 2 ⋅ 11)  =  4 ⋅ ⋅ 2√11  =  16√11

3√12  =  3√(2  2  3)  =  3 ⋅ 2√3  =  6√3

Then, we have

2√99 + 2√27 - 4√176 - 3√12  =  6√11 + 6√3 - 16√11 - 6√3

2√99 + 2√27 - 4√176 - 3√12  =  -10√11

Question 7 :

Simplify the following radical expression

3√16 - 3√2 + 43√54

Solution :

Decompose 16 and 54 into prime factors.

3√16  =  √(2  2  2 ⋅ 2)  =  23√2

43√54  =  43√(3  3  3 ⋅ 2)  =  4(33√2) = 123√2

Then, we have

3√16 - 3√2 - 43√54  =  23√2 - 3√2 + 123√2

3√16 + 3√2 - 43√54  =  133√2

Question 8 :

Simplify the following radical expression

3√24 + 3√375 - 3√3

Solution :

Decompose 24 and 375 into prime factors.

3√24  =  √(2  2  2 ⋅ 3)  =  23√3

3√375  =  3√(5  5  5 ⋅ 3)  =  53√3

Then, we have

3√24 + 3√375 - 3√3 = 23√3 + 53√3 - 3√3

3√24 + 3√375 - 3√3 = 63√3

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