When the denominator of an expression contains a term with a square root or a number within radical sign, the process of converting into an equivalent expression whose denominator is a rational number is called rationalizing the denominator.

Here we are going to some example problems to understand how to find the value of the variables by rationalizing the denominator.

**Example 1 : **

Rationalize the denominator (2 + √3)/(2 - √3) = x + y √3 and find the value of x and y.

**Solution : **

**Steps involved in the above solution :**

**Step 1 :**

Here we have 2 - √3 in the denominator, to rationalize the denominator we have multiply the entire fraction by its conjugate

Conjugate of 2 - √3 is 2 + √3.

**Step 2 :**

(i) By comparing the numerator (2 + √3)² with the algebraic identity (a+b)²=a²+ 2ab+b², we get 2² + 2(2)√3 + √3² ==> (7+4√3)

(ii) By comparing the denominator with the algebraic identity (a+b) (a-b) = a² - b², we get 2² - √3²

**Step 3 :**

By comparing this we get x = 7 and y = 4 as the final answer.

**Example 2 : **

Rationalize the denominator (5 + 4√3)/(4 + 5√3) = x + y √3 and find the value of x and y.

**Solution : **

**Steps involved in the above solution :**

**Step 1 :**

Here we have 4 + 5√3 in the denominator, to rationalize the denominator we have multiply the entire fraction by its conjugate

Conjugate of 4+5√3 is 4-5√3

**Step 2 :**

(i) In the numerator we have (5 + 4√3) (4-5√3). By multiplying these terms we get, 40 + 9√3

(ii) By comparing the numerator (2 + √3)² with the algebraic identity (a+b)²=a²+ 2ab+b², we get 4²-(5√3)² ==> -59

(iii) By cancelling the negative in numerator and denominator, we get

x = 40/59 and y = 9/59

**Example 3 : **

Rationalize the denominator (1+2√3)/(2-√3) = x+y√3 and find the value of x and y.

**Solution : **

**Steps involved in the above solution :**

**Step 1 :**

Here we have 2-√3 in the denominator, to rationalize the denominator we have multiply the entire fraction by its conjugate

Conjugate of 2-√3 is 2+√3

**Step 2 :**

(i) In the numerator we have (1+2√3) (2+√3). By multiplying these terms we get, 2 + 6 + 5√3

(ii) By comparing the denominator (2+√3)(2-√3) with the algebraic identity a²-b²=(a+b)(a-b), we get 2²-√3²==>1

By comparing this we get x = 8 and y = 5 as the final answer.

**Example 4 : **

Rationalize the denominator (3 + √5)/(3 - √5) + (3 - √5)/(3 + √5) = x + y √5 and find the value of x and y.

**Solution :**

**Steps involved in the above solution :**

**Step 1 :**

By taking L.C.M, we get (3 +√5)² + (3-√5)²/(3+√5)(3-√5)

**Step 2 :**

Expansion of (3+√5)² is 3²+2(3)(√5)+√5²

Expansion of (3-√5)² is 3²-2(3)(√5)+√5²

By comparing the denominator (3-√5)(3+√5) with the algebraic identity a²-b²=(a+b)(a-b), we get 3²-√5²==>4

**Step 3 :**

By comparing the L.H.S and R.H.S, we get x = 7 and y = 0

**Example 5 : **

Rationalize the denominator [(√5-√7)/(√5+√7)]-[(√5+√7)/ (√5 - √7)] = x + y √35 and find the value of x and y.

**Solution :**

**Steps involved in the above solution :**

**Step 1 :**

By taking L.C.M, we get

(√5-√7)²-(√5+√7)²/(√5+√7)(√5-√7)

**Step 2 :**

Expansion of (√5 - √7)² is

√5² + 2(√5)(√7) + √7²

Expansion of (√5+√7)²

√5² - 2(√5)(√7) + √7²

By comparing the denominator (√5 + √7)(√5 - √7) with the algebraic identity

a² - b² = (a + b)(a - b)

we get,

√5² - √7² = -2

**Step 3 :**

By combining the like terms we get 4√35/2

**Step 4 :**

By comparing the L.H.S and R.H.S we get the values of x and y

x = 0 and y = 2

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