Rationalization of Surds –
Suppose we have taken a number whose denominator is irrational. Then, the process of converting its denominator to a rational number, by multiplying it’s numerator and denominator by a suitable number, is called rationalization.
Rationalizing Factor (RF) –
If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other, abbreviated as RF.
If ‘a’ and ‘b’ are integers and ‘x’, ‘y’, are natural numbers, then
1) (a + √b) and (a - √b) are RF of each other, as
(a + √b) X (a - √b) = (a² - √b²) = (a² - b), which is rational.
2) (a + b√x) and (a - b√x) are RF of each other, as
(a + b√x) X (a - b√x) = (a² - b²√x²) = (a² - b²x), which is rational.
3) (√x + √y) and (√x - √y) are RF of each other, as
(√x + √y) X (√x - √y) = (√x²- √y²) = (x – y), which is rational.
9
Example.1) Simplify ------ by rationalizing the denominator.
√3
Ans.) On multiplying the numerator and denominator of the given number by √3, we get as follows –
9 9 X √3 9√3
------- = ---------- = ------- = 3√3 (Ans.)
√3 √3 X √3 3
14
Example.2) Simplify -------- by rationalizing the denominator.
(3 + √2)
Ans.) On multiplying the numerator and denominator of the given number by (3 - √2), we get as follows
14 14 X (3 - √2) 14 X (3 - √2)
---------- = -------------------- = -----------------
(3 + √2) (3 + √2) X (3 - √2) (3² - √2²)
14 X (3 - √2)
= ---------------- = 2 (3 - √2) (Ans.)
9 - 2