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  =  2√5

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

³√72  =  ³√(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3)

³√72  =  2 ⋅ ³√(3 ⋅ 3)

³√72  =  2 ⋅ ³√(3 ⋅ 3)

³√72  =  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

⁵√288  =  ⁵√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)

⁵√288  =  2 ⁵√3

Question 10 : 

Simplify : 

⁶√320

Solution : 

Decompose 320 into prime factors using synthetic division.

So, we have

⁶√320  =  ⁶√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5)

⁶√320  =  2⁶√5

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