SIMPLIFYING SQUARE ROOT EXPRESSIONS WITH VARIABLES

Key Concept

To simplify the square root expressions, write each term inside the radical as squares. 

We can get one term out of the square root for every two same terms multiplied inside the radical. 

Solved Examples

Example 1 :

Simplify :

√(16u⁴v³)

Solution : 

=  √(16u⁴v³)

=  √(4² ⋅ u² ⋅ u² ⋅ v² ⋅ v)

=  (4 ⋅ u ⋅ u ⋅ v)√v

=  4u²v√v

Example 2 :

Simplify :

√(147m³n³)

Solution : 

=  √(147m³n³)

=  √(3 ⋅ 7² ⋅ m² ⋅ m ⋅ n² ⋅ n)

=  (7 ⋅ m ⋅ n)√(3mn)

=  7mn√(3mn)

Example 3 :

Simplify :

√(75x²y)

Solution : 

=  √(75x²y)

=  √(3 ⋅ 5² ⋅ x² ⋅ y)

=  (5 ⋅ x)√(3y)

=  5x√(3y)

Example 4 :

Simplify :

6√(72x²)

Solution : 

=  6√(72x²)

=  6√(2 ⋅ 6² ⋅ x²)

=  (6 ⋅ 6 ⋅ x)√2

=  36x√2

Example 5 :

Simplify :

√(x² + 2xy + y²)

Solution :

=  √(x² + 2xy + y²)

Use algebraic identity (a + b)²  =  a² + 2ab + b².

=  √(x + y)²

=  x + y

Example 6 :

Simplify :

√(p² - 2pq + q²)

Solution :

=  √(p² - 2pq + q²)

Use algebraic identity (a - b)²  =  a² - 2ab + b².

=  √(p - q)²

=  p - q

Example 7 :

Simplify :

√[(x² - y²)(x + y) / (x - y)]

Solution :

=  √[(x² - y²)(x + y) / (x - y)]

Use algebraic identity a² - b²  =  (a + b)(a - b).

=  √[(x + y)(x - y)(x + y) / (x - y)]

=  √[(x + y)(x + y)]

=  x + y

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