SIMPLIFYING RADICAL EXPRESSIONS

The following steps will be useful to simplify any radical expressions. 

Step 1 :

Decompose the number inside the radical into prime factors. 

Step 2 :

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

Step 3 :

If you have cube root (³√), you have to take one term out of cube root for every three same terms multiplied inside the radical.

Step 4 :

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

Step 5 :

Combine the radical terms using mathematical operations. 

Example : 

√18 + √8  =  √(3 ⋅ 3 ⋅ 2) + √(2 ⋅ 2 ⋅ 2)

 √18 + √8  =  3√2 + 2√2

 √18 + √8  =  5√2

Solved Examples 

Example 1 : 

Simplify the radical expression : 

√169 + √121

Solution : 

Decompose 169 and 121 into prime factors using synthetic division. 

So, we have

√169 + √121  =  13 + 11

√169 + √121  =  24

Example 2 : 

Simplify the radical expression : 

√20 + √320

Solution : 

Decompose 20 and 320 into prime factors using synthetic division. 

So, we have

√20 + √320  =  2√5 + 8√5

√20 + √320  =  10√5

Example 3 : 

Simplify the radical expression : 

√117 - √52

Solution : 

Decompose 117 and 52 into prime factors using synthetic division. 

So, we have

√117 - √52  =  3√13 - 2√13

√117 + √52  =  √13

Example 4 : 

Simplify the radical expression : 

√243 - 5√12 + √27 

Solution : 

Decompose 243, 12 and 27 into prime factors using synthetic division. 

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)  =  9√3

√12  =  √(2 ⋅ 2 ⋅ 3)  =  2√3

√27  =  √(3 ⋅ 3 ⋅ 3)  =  3√3

So, we have

√243 - 5√12 + √27  =  9√3 - 5(2√3) + 3√3

Simplify.

√243 - 5√12 + √27  =  9√3 - 10√3 + 3√3

√243 - 5√12 + √27  =  2√3

Example 5 : 

Simplify the radical expression : 

-√147 - √243 

Solution : 

Decompose 147 and 243 into prime factors using synthetic division. 

√147  =  √(7 ⋅ 7 ⋅ 3)  =  7√3

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)  =  9√3

So, we have

-√147 - √243  =  -7√3 - 9√3

-√147 - √243  =  -16√3

Example 6 : 

Simplify the radical expression : 

(√13)(√26)

Solution : 

Decompose 13 and 26 into prime factors. 

13 is a prime number. So, it can't be decomposed anymore.

√26  =  √(2 ⋅ 13)  =  √2 ⋅ √13

So, we have

(√13)(√26)  =  (√13)(√2 ⋅ √13)

(√13)(√26)  =  (√13 ⋅ √13)√2

(√13)(√26)  =  13√2

Example 7 : 

Simplify the radical expression : 

(3√14)(√35)

Solution : 

Decompose 14 and 35 into prime factors.

√14  =  √(2 ⋅ 7)  =  √2 ⋅ √7

√35  =  √(5 ⋅ 7)  =  √5 ⋅ √7

So, we have

(3√14)(√35)  =  3( √2 ⋅ √7)(√5 ⋅ √7)

(3√14)(√35)  =  3(√7 ⋅ √7)(√2 ⋅ √5)

(3√14)(√35)  =  3(7)√(2 ⋅ 5)

(3√14)(√35)  =  21√10

Example 8 : 

Simplify the radical expression : 

(8√117) ÷ (2√52)

Solution : 

Decompose 117 and 52 into prime factors using synthetic division.

(8√117) ÷ (2√52)  =  8(3√13) ÷ 2(2√13)

(8√117) ÷ (2√52)  =  24√13 ÷ 4√13

(8√117) ÷ (2√52)  =  24√13 / 4√13

(8√117) ÷ (2√52)  =  6

Example 9 : 

Simplify the radical expression : 

(8√3)²

Solution :

(8√3)²  =  8√3 ⋅ 8√3

(8√3)²  =  (8 ⋅ 8)(√3 ⋅ √3)

(8√3)²  =  (64)(3)

(8√3)²  =  192

Example 10 : 

Simplify the radical expression : 

(√2)³ + √8

Solution :

(√2)³ + √8  =  (√2 ⋅ √2 ⋅ √2) + √(2⋅ 2 ⋅ 2)

(√2)³ + √8  =  (2 ⋅ √2) + 2√2

(√2)³ + √8  =  2√2 + 2√2

(√2)³ + √8  =  4√2

Example 11 :

Simplify : 

⁴√(x⁴/16)

Solution :

⁴√(x⁴/16)  =  ⁴√(x⁴) / ⁴√16

⁴√(x⁴/16)  =  ⁴√(x ⋅ x ⋅ x ⋅ x) / ⁴√(2 ⋅ 2 ⋅ 2 ⋅ 2)

⁴√(x⁴/16)  =  x / 2

Example 12 :

Simplify : 

³√(125p⁶q³)

Solution :

³√(125p⁶q³)  =  ³√(5 ⋅ 5 ⋅ 5 ⋅ p² ⋅ p² ⋅ p² ⋅ q ⋅ q ⋅ q)

³√(125p⁶q³)  =  5p²q

Example 13 :

If √(0.9 ⋅ 0.09 ⋅ x) = 0.9 ⋅ 0.9√3, then the value of x/3 is :

Solution :

√(0.9 ⋅ 0.09 ⋅ x) = 0.9 ⋅ 0.9√3

Squaring on both sides

(√(0.9 ⋅ 0.09 ⋅ x))² = (0.9 ⋅ 0.9)² (√3)²

(0.9 ⋅ 0.09 ⋅ x) = (0.9 ⋅ 0.9)² 3

x/3 =  (0.9 ⋅ 0.9)² / (0.9 ⋅ 0.09)

x/3 = 0.81/(0.9 ⋅ 0.09)

x/3 = 10

Example 14 :

Find the value of (√1521/11) ⋅ (11/√196)

Solution :

= (√1521/11) ⋅ (11/√196)

By cancelling the numerator and denominator, we get

= (√1521/√196)

= √(39⋅39)/√(14 ⋅ 14)

= 39/14

Example 15 :

Find the value of [ √(7√7√7√7) ]

Solution :

=  [ √(7√(7√(7√(7))) ]

Here inside a square root, we have square root of 7 as four times.

= [7 [7 [7 (7)^1/2]^1/2]^1/2]^1/2

= [7 [7 [(7)^3/2]^1/2]^1/2]^1/2

= [7 [7 (7)^3/4]^1/2]^1/2

= [7 [7^{1 + 3/4}]^1/2]^1/2

= [7 [7^7/4]^1/2]^1/2

= [7 ⋅ 7^7/8]]^1/2

= [7 ^{1 + 7/8}]^1/2

= [7 ^15/8]^1/2

= 7 ^15/16

Example 16 :

9 √x = √12 + √147, then x is

Solution :

9 √x = √12 + √147

Doing possible simplification, we get

9 √x = √(2⋅2⋅3) + √(3⋅7⋅7)

9 √x = 2√3 + 7√3

9 √x = 9√3

By comparing the corresponding terms, we get

√x = √3

x = 3

So, the value of x is 3.

Example 17 :

Find the value of √2304 + √23.04 + √0.2304

Solution :

= √2304 + √23.04 + √0.2304

√2304 = √(48 ⋅ 48)

= 48

√23.04 = √2304/100

= √(48 ⋅ 48)/(10 ⋅ 10)

= 48/10

= 4.8

 √0.2304 = √(2304/10000)

= √(48⋅48)/(100⋅100)

= 48/100

= 0.48

 √2304 + √23.04 + √0.2304 = 48 + 4.8 + 0.48

= 53.28

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