When a radical contains an expression that is not a perfect root, for example, the square root of 3 or cube root of 5, it is called an irrational number.

So, in order to rationalize the denominator, we have to get rid of all radicals that are in denominator.

The following steps are involved in rationalizing the denominator of rational expression.

**Step 1 : **

Multiply both numerator and denominator by a radical that will get rid of the radical in the denominator.

If the radical in the denominator is a square root, then we have to multiply by a square root that will give us a perfect square under the radical when multiplied by the denominator.

**Step 2 : **

Make sure all radicals are simplified.

Some radicals will already be in a simplified form, but we have to make sure that we simplify the ones that are not.

**Step 3 : **

Simplify the fraction if needed.

Be careful. We cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

**Example 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

**Example 2 : **

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

**Example 3 : **

Rationalize the denominator :

5 / √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

**Example 4 : **

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

**Example 5 : **

Rationalize the denominator :

1 / (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 a^{2} - b^{2} = (a + b)(a - b), simplify the denominator on the right side.

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

1 / (3 + √2) = (3-√2) / (9 - 2)

1 / (3 + √2) = (3 - √2) / 7

**Example 6 : **

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] / [3^{2} - (√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

**Example 7 : **

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^{2 }- (√y)^{2}]

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

**Example 8 : **

Rationalize the denominator :

^{3}√(2/3a)

**Solution :**

^{3}√(2/3a) = ^{3}√2 / ^{3}√3a

To rationalize the denominator in this case, multiply both numerator and denominator on the right side by the cube root of 9a^{2}.

^{3}√(2/3a) = [^{3}√2 ⋅ ^{3}√(9a^{2})] / [^{3}√3a ⋅ ^{3}√(9a^{2})]

Simplify.

^{3}√(2/3a) = ^{3}√(18a^{2}) / ^{3}√(27a^{3})

^{3}√(2/3a) = ^{3}√(18a^{2}) / ^{3}√(3 ⋅ 3 ⋅ 3 ⋅ a ⋅ a ⋅ a)

^{3}√(2/3a) = ^{3}√(18a^{2}) / 3a

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