**Case 1 :**

If the denominator is in the form of √b (where b is a rational number), then we have to multiply both the numerator and denominator by the same √b to rationalize the denominator.

**Case 2 :**

If the denominator is in the form of a ± √b or a ± c√b (where b is a rational number), then we have to multiply both the numerator and denominator by its conjugate.

a + √b and a - √b are conjugate to each other

a + c√b and a - c√b are conjugate to each other

**Case 3 :**

If the denominator is in the form of √a ± √b (where a and b are rational numbers), then we have to multiply both the numerator and denominator by its conjugate.

√a + √b and √a - √b are conjugate to each other

**Example 1 :**

Simplify :

18 / √6

**Solution :**

Simplifying the above radical expression is nothing but rationalizing the denominator.

So, rationalize the denominator.

**Here, the denominator is **√6.

In the given fraction, multiply both numerator and denominator by √6.

18 / √6 = (18√6) / (√6 ⋅ √6)

18 / √6 = 18√6 / 6

18 / √6 = 3√6

**Example 2 :**

Simplify :

1 / (2 + √5)

**Solution :**

Simplifying the above radical expression is nothing but rationalizing the denominator.

So, rationalize the denominator.

**Here, the denominator is 2 + **√5.

In the given fraction, multiply both numerator and denominator by the conjugate of **2 + **√5. That is **2 - **√5.

**Example 3 :**

Simplify :

(6 + √5) / (6 - √5)

**Solution :**

Simplifying the above radical expression is nothing but rationalizing the denominator.

So, rationalize the denominator.

**Here, the denominator is 6 - **√5.

In the given fraction, multiply both numerator and denominator by the conjugate of 6** - **√5. That is 6** + **√5.

(6 + √5) / (6 - √5) = [(6+√5)(6+√5)] / [(6-√5)(6+√5)]

(6 + √5) / (6 - √5) = [(6+√5)(6+√5)] / [(6-√5)(6+√5)]

(6 + √5) / (6 - √5) = (6 + √5)^{2} / [6^{2 }- (√5)^{2}]

(6 + √5) / (6 - √5) = [6^{2} + 2(6)(√5) + (√5)^{2}] / (36^{ }- 5)

(6 + √5) / (6 - √5) = [36 + 12√5 + 5] / 31

(6 + √5) / (6 - √5) = (41 + 12√5) / 31

**Example 4 :**

Find the values of 'x' and 'y' :

(2 + √3)/(2 - √3) = x + y√3

**Solution : **

**(2 + √3)/(2 - √3) = x + y√3**

**On the left side of the above equation, multiply both numerator and denominator by the conjugate of **2** - **√3. That is 2** + **√3.

[(2+√3)(2+√5)] / [(2-√3)(2+√3)] = **x + y√3**

(2 + √3)^{2} / [2^{2 }- (√3)^{2}] = **x + y√3**

[2^{2} + 2(2)(√3) + (√3)^{2}] / (4^{ }- 3) = **x + y√3**

**[4 + 4√3 + 3] / 1 = x + y√3**

**7 + 4√3 = x + y√3**

**Therefore, **

**x = 7**

**y = 4**

**Example 5 :**

Simplify :

√(12x^{2}) / √(30x)

**Solution : **

√(12x^{2}) / √(30x) = √(12x^{2}/30x)

√(12x^{2}) / √(30x) = √(2x/5)

√(12x^{2}) / √(30x) = √(2x) / √5

On the right side, multiply both numerator and denominator by √5.

√(12x^{2}) / √(30x) = √(2x) ⋅ √5 / √5 ⋅ √5

√(12x^{2}) / √(30x) = √(2x ⋅ 5) / 5

√(12x^{2}) / √(30x) = √10x / 5

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