# RATIONALIZING THE DENOMINATOR WORKSHEET

Problem 1 :

Rationalize the denominator :

12 / √6

Problem 2 :

Rationalize the denominator :

√18 / (3√2)

Problem 3 :

Rationalize the denominator :

4√5 / √10

Problem 4 :

Rationalize the denominator :

/ √7

Problem 5 :

Rationalize the denominator :

12 / √72

Problem 6 :

Rationalize the denominator :

(3 - √3) / √3

Problem 7 :

Rationalize the denominator :

/ (3 + √2)

Problem 8 :

Rationalize the denominator :

(1 - √5) / (3 + √5)

Problem 9 :

Rationalize the denominator :

(√x + y) /  (x - √y)

Problem 10 :

Rationalize the denominator :

3√(2/3a)

Problem 1 :

Rationalize the denominator :

12 / √6

Solution :

Multiply both numerator and denominator by √6 to get rid of the radical in the denominator.

12 / √6  =  (12 ⋅ √6) / (√6 ⋅ √6)

12 / √6  =  12√6 / 6

12 / √6  =  2√6

Problem 2 :

Rationalize the denominator :

√18 / (3√2)

Solution :

Simplify.

√18 / (3√2)  =  √(3 ⋅ 3 ⋅ 2) / (3√2)

√18 / (3√2)  =  3√2 / (3√2)

√18 / (3√2)  =  1

Problem 3 :

Rationalize the denominator :

4√5 / √10

Solution :

Simplify.

4√5/√10  =  4√5 / √(2 5)

4√5/√10  =  4√5 / (√2 ⋅ √5)

On the right side, cancel out √5 in numerator and denominator.

4√5/√10  =  4 / √2

On the right side, multiply both numerator and denominator by √2 to get rid of the radical in the denominator.

4√5/√10  =  (4 ⋅ √2) / (√2 ⋅ √2)

4√5/√10  =  4√2 / 2

4√5/√10  =  2√2

Problem 4 :

Rationalize the denominator :

/ √7

Solution :

Multiply both numerator and denominator by √7 to get rid of the radical in the denominator.

5 / √7  =  (5 ⋅ √7) / (√7 ⋅ √7)

5 / √7  =  5√7 / 7

Problem 5 :

Rationalize the denominator :

12 / √72

Solution :

Decompose 72 into prime factor using synthetic division.

√72  =  √(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3)

√72  =  2 ⋅ 3 ⋅ √2

√72  =  6√2

Then, we have

12 / √72  =  12 / 6√2

Simplify.

12 / √72  =  2 / √2

On the right side, multiply both numerator and denominator by √2 to get rid of the radical in the denominator.

12 / √72  =  (2 ⋅ √2) ⋅ (√2 ⋅ √2)

12 / √72  =  2√2 / 2

12 / √72  =  √2

Problem 6 :

Rationalize the denominator :

(3 - √3) / √3

Solution :

To get rid of the radical in denominator, multiply both numerator and denominator by √3.

(3 - √3) / √3  =  [(3-√3) ⋅ √3] / (√3 ⋅ √3)

(3 - √3) / √3  =  (3√3 - 3) / 3

(3 - √3) / √3  =  3(√3 - 1) / 3

(3 - √3) / √3  =  √3 - 1

Problem 7 :

Rationalize the denominator :

/ (3 + √2)

Solution :

To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √2), that is by (3 - √2).

1 / (3 + √2)  =  [1 ⋅ (3-√2)] / [(3+√2)  (3-√2)]

1 / (3 + √2)  =  (3-√2) / [(3+√2)  (3-√2)]

Using the algebraic identity a2 - b2  =  (a + b)(a - b), simplify the denominator on the right side.

1 / (3 + √2)  =  (3-√2) / [32 - (√2)2]

1 / (3 + √2)  =  (3-√2) / (9 - 2)

1 / (3 + √2)  =  (3 - √2) / 7

Problem 8 :

Rationalize the denominator :

(1 - √5) / (3 + √5)

Solution :

To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √5), that is by (3 - √5).

(1 - √5) / (3 + √5)  =  [(1-√5) ⋅ (3-√5)] / [(3+√5)  (3-√5)]

Simplify.

(1 - √5) / (3 + √5)  =  [3 √5 - 3√5 + 5] / [32 - (√5)2]

(1 - √5) / (3 + √5)  =  (8 - 4√5) / (9 - 5)

(1 - √5) / (3 + √5)  =  4(2 - √5) / 4

(1 - √5) / (3 + √5)  =  2 - √5

Problem 9 :

Rationalize the denominator :

(√x + y) /  (x - √y)

Solution :

To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (x - √y), that is by (x + √y).

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

Simplify.

(√x + y) / (x - √y)  =  [x√x + √xy + xy + y√y] / [(x- (y)2]

(√x + y) / (x - √y)  =  [x√x + √xy + xy + y√y] / (x- y2)

Problem 10 :

Rationalize the denominator :

3√(2/3a)

Solution :

3√(2/3a)  =  3√2 / 33a

To rationalize the denominator in this case, multiply both numerator and denominator on the right side by the cube root of 9a2.

3√(2/3a)  =  [3√2 ⋅ 3√(9a2)] / [3√3a ⋅ 3√(9a2)]

Simplify.

3√(2/3a)  =  3√(18a2) / 3√(27a3)

3√(2/3a)  =  3√(18a2) / 3√(3 ⋅ 3 ⋅ 3 ⋅ a ⋅ a ⋅ a)

3√(2/3a)  =  3√(18a2) / 3a

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