## RATIONALIZATION OF SURDS

Rationalization of surds :

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

If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other.

## How to rationalize the denominator

Case 1 :

If the denominator is in the form of √a (where a is a rational number).

Then we have to multiply both the numerator and denominator by the same (√a).

Example 1 :

Rationalize the denominator 18/√6

Solution :

Step 1:

We have to rationalize the denominator. Here we have √6  (in the form of √a). Then we have to multiply the numerator and denominator by √6

Step 2 :

By multiplying the numerators and denominators of first and second fraction , we get

Step 3 :

By simplifications, we get 3√6

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 of each other.

a + c√b and a - c√b are conjugate of each other.

Example 2 :

Rationalize the denominator 4/(1+2√3)

Solution :

Step 1:

Here we have (1+2√3)  (in the form of a + c√b) in the denominator. Then we have to multiply the numerator and denominator by the conjugate of  (1+2√3).

Conjugate of (1 + 2√3) is (1 - 2√3)

Step 2 :

By multiplying the numerators and denominators of first and second fraction, we get

Step 3 :

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

Case 3 :

If the denominator is in the form of ± √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 of each other.

Example 3 :

Rationalize the denominator (6 + √5)/(6-√5)

Solution :

Step 1:

Here we have (6-√5) in the denominator. Then we have to multiply the numerator and denominator by the conjugate of  (6-√5).

Conjugate of (6-√5) is (6+√5)

## Rationalizing the denominator with variables

Example 4 :

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

Solution :

Now we have to compare the final answer with R.H.S

The values of x and y are 7 and 4 respectively.

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