RATIONALIZING THE DENOMINATOR WITH VARIABLES

Rationalize the denominator in the following examples.

Example 1 : 

¹⁄√ₓ

Solution : 

= ¹⁄√ₓ

Multiply both the numerator and denominator by √x. 

=  (1 ⋅ √x)/(√x ⋅ √x)

=  √x/x

Example 2 : 

¹⁄₍ₓ ₊ √y₎

Solution : 

= ¹⁄₍ₓ ₊ √y₎

Multiply both numerator and denominator by (x - √y).

=  [1 ⋅ (x - √y)] / [(x + √y)(x - √y)]

Use the algebraic identity a² - b² = (a + b)(a - b) in denominator to simplify.

=  (x - √y) / [x² - (√y)²]

=  (x - √y) / (x² - y)

Example 3 : 

⁽^√ˣ ⁺ ^√ʸ⁾⁄√x

Solution : 

=  ⁽^√ˣ ⁺ ^√ʸ⁾⁄√x

Multiply both the numerator and denominator by √x. 

=  (√x + √y)√x / (√x ⋅ √x)

Distribute and simplify. 

=  [(√x ⋅ √x) + (√y ⋅ √x)] / x

=  [x + √(xy)]/x

Example 4 : 

(√x + √y)/(√x - √y)

Solution : 

=  (√x + √y)/(√x - √y)

Multiply both numerator and denominator by (x + √y).

=  [(√x + √y)(√x + √y)] / [(√x - √y)(√x + √y)]

=  (√x + √y)² / [(√x)² - (√y)²]

=  [(√x)² + 2√x√y + (√y)²] / (x - y)

= (x + 2√(xy) + y) / (x - y)

Example 5 : 

√(100x/11y)

Solution : 

=  √(100x/11y)

Distribute the radical to numerator and denominator. 

=  √(100x)/√(11y)

So, multiply both numerator and denominator by the 11y. 

=  [√(100x) ⋅ √(11y)] / √(11y) ⋅ √(11y)]

Simplify.

=  √(100x ⋅ 11y) / 11y

100 is a perfect square and √100 = 10.

=  10√(11xy) / 11y

Example 6 : 

Find the value of ab.

1/(x + y√3)

Solution : 

=  1/(x + y√3)

Multiply both numerator and denominator by (x - y√3). 

=  [1 ⋅ (x - y√3)] / [(x + y√3)(x - y√3)]

Use the algebraic identity a² - b² = (a + b)(a - b) in denominator to simplify.

=  (x - y√3) / [x² - (y√3)²]

=  (x - y√3) / [x² + y²(√3)²]

=  (x - y√3) / (x² + 3y²)

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