## SIMPLIFY RADICAL EXPRESSIONS USING THE DISTRIBUTIVE PROPERTY

Simplify radical expressions using the distributive property :

The method of multiplying the radical terms which are added or subtracted inside the bracket by the common radical term or numerical value outside the bracket is know as distributive property. By using the distributive property of radicals, we can show the answer in the simplest form.

Steps involved in simplifying radicals : ## Simplify radical expressions using distributive property - Examples

Example 1 :

Simplify the radical expression given below

√3(√3 + √12)

Solution : Example 2 :

Simplify the radical expression given below

-√2(-4 -√2 )

Solution :

=  -√2(-4 -√2 )

Distribute -√2 inside the bracket.

=  -√2(-4) -√2 (-√2 )

For every two same numbers which are multiplying inside the radical sign, we can take one number commonly from the radical.

=  4√2 + 2

Hence the answer is 4√2 + 2.

Example 3 :

Simplify the radical expression given below

√2(7 + √5 )

Solution :

=  √2(7 + √5 )

Distribute √2 inside the bracket.

=  √2(7) + √2 (√5 )

=  7√2 + √(2x5)

=  7√2 + √10

Hence the answer is 7√2 + √10

Example 4 :

Simplify the radical expression given below

2(√4 + √10)

Solution :

=  2(√4 + √10)

=  2 (√4) + 2 (√10)

=  2 (√(2 x 2) + 2 √10

=  (2 x 2) + 2 √10

=  4 + 2 √10

Hence the answer is 4 + 2 √10

Example 5 :

Simplify the radical expression given below

√5(√8 + √10)

Solution :

=  √5(√8 + √10)

=  √5 (√8) + √5 (√10)

=  √(5 x 8) + √(5 x 10)

=  (√(5 x 2 x 2 x 2) + √(5 x 5 x 2)

=  2(√(5 x 2) +  5√2

= 2√10 +  5√2

=  4 + 2 √10

Hence the answer is 4 + 2 √10

Example 6 :

Simplify the radical expression given below

√3(√9 + √21)

Solution :

= √3(√9 + √21)

=  √3 (√9) + √3 (√21)

=  √(3 x 9) + √(3 x 21)

=  (√(3 x 3 x 3) + √(3 x 3 x 7)

=  3√3 +  3√7

Hence the answer is 3√3 +  3√7

Example 7 :

Simplify the radical expression given below

2√5(√6 + 2)

Solution :

=  2√5(√6 + 2)

=  2√5 (√6) + 2√5(2)

=  2√(5 x 6) + 2√(5 x 2)

=  2√30 + 2√10

Hence the answer is 2√30 + 2√10

Example 8 :

Simplify the radical expression given below

√14x(3 - √2x)

Solution :

=  √14x(3 - √2x)

=  √14x (3) - √14x(√2x)

=  √42x  - √(14 ⋅ 2 ⋅ x ⋅ x)

=  √42x  - √(2 ⋅ 7 ⋅ 2 ⋅ x ⋅ x)

=  √42x  - 2x√7

Hence the answer is √42x  - 2x√7

Example 9 :

Simplify the radical expression given below

√21x(5 + √7)

Solution :

=  √21x(5 + √7)

=  √21x (5) + √21x (√7)

=  5√21x  + √(21 ⋅ x ⋅ 7)

=  5√21x  + √(3 ⋅ 7 ⋅ x ⋅ 7)

=  5√21x  + 7√3x

Hence the answer is  5√21x  + 7√3x

Example 10 :

Simplify the radical expression given below

(5 - √3(5 + √3)

Solution :

=  (5 - √3(5 + √3)

=  5(5) + 5√3 - 5√3 - √3√3

=  25 - √(3 ⋅ 3)

=  25 - 3

=  22

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