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.

The following steps will be useful to simply radical expressions

Step 1 :

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 :

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 :

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 :

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 :

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 :

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 :

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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