SIMPLIFYING RADICALS

Thew following steps will be useful to simplify radicals. 

Step 1 :

Decompose the number inside the radical sign into prime factors.  

Step 2 :

If you have square root (√), you have to take one number out of the square root for every two same numbers multiplied inside the radical.

If you have cube root (3), we have to take one number out of cube root for every three same numbers multiplied inside the radical.

If you have fourth root (4), you have to take one number out of fourth root for every four same numbers multiplied inside the radical.

Step 3 :

Simplify. 

Examples :

Solved Questions

Question 1 : 

Simplify : 

√20

Solution : 

Decompose 20 into prime factors using synthetic division. 

So, we have

√20  =  √(2 ⋅ 2 ⋅ 5)

√20  =  25

Question 2 : 

Simplify : 

√121

Solution : 

Decompose 121 into prime factors.

√121  =  √(11 ⋅ 11)

√121  =  11

Question 3 : 

Simplify : 

√52

Solution : 

Decompose 52 into prime factors using synthetic division.

So, we have

√52  =  √(2 ⋅ 2 ⋅ 13)

√52  =  2√13

Question 4 : 

Simplify : 

√45

Solution : 

Decompose 45 into prime factors using synthetic division.

So, we have

√45  =  √(5 ⋅ 3 ⋅ 3)

√45  =  3√5

Question 5 : 

Simplify : 

3√72

Solution : 

Decompose 72 into prime factors using synthetic division.

So, we have

372  =  3(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3)

372  =  2 ⋅ 3√(⋅ 3)

372  =  2 ⋅ 3√(⋅ 3)

372  =  23√9

Question 6 : 

Simplify : 

 

3√40

Solution : 

Decompose 40 into prime factors using synthetic division.

So, we have

3√40  =  3(2 ⋅ 2 ⋅ 2 ⋅ 5)

3√40  =  23√5

Question 7 : 

Simplify : 

3√27

Solution : 

Decompose 27 into prime factors using synthetic division.

So, we have

3√27  =  3(3 ⋅ 3 ⋅ 3)

3√27  =  3

Question 8 : 

Simplify : 

 

4√243

Solution : 

Decompose 243 into prime factors using synthetic division.

So, we have

4√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

4√243  =  34√3

Question 9 : 

Simplify : 

5√288

Solution : 

Decompose 288 into prime factors using synthetic division.

So, we have

5288  =  5(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)

5288  =  2 5√3

Question 10 : 

Simplify : 

6√320

Solution : 

Decompose 320 into prime factors using synthetic division.

So, we have

6320  =  6(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5)

6320  =  26√5

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