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

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

Step 1 :

Step 2 :

Take one number out of the radical for every two same numbers multiplied inside the radical sign.

Step 3 :

Simplify.

Examples : ## Simplifying Radical Expressions - Examples

Example 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)

Example 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

Example 3 :

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

Example 4 :

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

Example 5 :

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)

Example 6 :

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

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