SIMPLIFYING RADICAL EXPRESSIONS

About "Simplifying Radical Expressions"

Simplifying Radical Expressions :

In this section, we are going to learn how to simplify radical expressions. 

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

(i)  Decompose the number inside the radical into prime factors. 

(ii)  Inside the radical, if the same number is repeated twice with multiplication, it can be taken out of the radical.

(iii) 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

Simplifying Radical Expressions - 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

After having gone through the stuff given above, we hope that the students would have understood, "Simplifying Radical Expressions". 

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