# SIMPLIFYING SQUARE ROOT EXPRESSIONS WITH VARIABLES

## About "Simplifying square root expressions with variables"

Simplifying square root expressions with variables :

Here we are going to see how to simplify square root expressions with variables.

The following examples will illustrate the method of simplifying square root expression with variables.

Example 1 :

Simplify (10 + √3) (2 + √5)

Solution :

(10 + √3) (2 + √5)

To find the product of the given expressions, we have to use the distributive property.

=  (10 + √3) (2 + √5)

=  10 (2) + 10 √5 + √3 (2) + 3 (√5)

=  20 + 10 √5 + 2√3 + √(3 ⋅ 5)

=  20 + 10 √5 + 2√3 + √15

No two terms are not having same radicand, so we cannot combine the terms.

Hence the answer is 20 + 10 √5 + 2√3 + √15.

Example 2 :

Simplify (√5 + √3)2

Solution :

(√5 + √3)2

The given expression exactly matches with the algebraic identity (a + b)2

(a + b) =  a2 + 2ab + b2

a  =  √5  and b  =  √3

(√5 + √3) =  (√5)2 + 2 (√5)(√3) + (√3)2

=  5 + 2 √(5 ⋅ 3) + 3

=  5 + 3  + 2 √15

=  8 + 2 √15

Hence the answer is 8 + 2 √15.

Example 3 :

Simplify (√13 - √2)(√13 + √2)

Solution :

(√13 - √2)(√13 + √2)

The given expression exactly matches with the algebraic identity (a + b)(a - b)

(a + b)(a - b)  =  a2 - b2

(√13 - √2)(√13 + √2)  =  (√13)2 - (√2)2

=  13 - 12

=  1

Example 4 :

Simplify (8 + √3)(8 - √3)

Solution :

(8 + √3)(8 - √3)

The given expression exactly matches with the algebraic identity (a + b)(a - b)

(a + b)(a - b)  =  a2 - b2

(8 + √3)(8 - √3)  =  (8)2 - (√3)2

=  64 - 3

=  61

Example 5 :

Simplify (5 √3)(8 - 2√5)

Solution :

(5 √3)(8 - 2√5)

To find the product of the given expressions, we have to use the distributive property.

=  (5 √3)(8 - 2√5)

=  5 (8) + 5 (-2√5) + √3 (8) + 3 (-2√5)

=  40 - 10 √5 + 8√3 - 2√(3 ⋅ 5)

=  40 - 10 √5 + 8√3 - 2√15

No two terms are not having same radicand, so we cannot combine the terms.

Hence the answer is 40 - 10 √5 + 8√3 - 2√15.

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