RATIONALIZING THE DENOMINATOR WITH VARIABLES WORKSHEET

Rationalize the denominator in the following expressions :

1)  1/√x

2)  1/(x + √y)

3)  (√x + √y)/√x

4)  (√x + √y)/(√x - √y)

5)  √(100x/11y)

6)  1/(x + y√3)

7)  √12x²/√30x

8)  √81p² / √45p²

9)  2/√6r²

10) Find the area and perimeter of the trapezoid. (Hint: The area of a trapezoid is the product of half the height 6√3 meters and the sum of the bases 2√63 and 7√27 meters.)

Detailed Answers 

1. Answer : 

=  1/√x

Multiply both the numerator and denominator by √x. 

=  (1 ⋅ √x)/(√x ⋅ √x)

=  √x/x

2. Answer : 

=  1/(x + √y)

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

Multiply both the numerator and denominator by √x. 

=  [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)

3. Answer : 

=  (√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

4. Answer : 

=  (√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)

5. Answer : 

=  √(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

6. Answer : 

=  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)

7. Answer :

= √12x²/√30x

Rationalizing the denominator, we get

= (√12x²/√30x) ⋅ (√30x/√30x)

= (√12x²√30x) / (√30x ⋅ √30x)

= √(12x² ⋅ 30x) / √(30x ⋅ √30x)

= √(6 ⋅ 2x² ⋅ 6 ⋅ 5x) / √(30x ⋅ 30x)

= 6x√10x / 30x

= √10x / 5

8. Answer :

=  √81p² / √45p²

=  (√81p²  √45p²) / (√45p²  ⋅  √45p²)

=  (√81p² 45p) / √(45p²  ⋅ 45p²)

=  (√9  ⋅ 9p p)/ / (45p²) 45p²  ⋅ 45p)²

9. Answer :

= 2/√6r²

= (2/√6r²) ⋅ (√6r²/√6r²)

= (2√6r²)

/√6r²) ⋅ (/²√6r)²

9) 

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