Rationalization
There is a irrational number suppose a + √b, let us multiply a - √b, and the product of these two irrational numbers would be a rational number. (a + √b) . (a - √b) = a² - (√b)² = a² – b, this process is called Rationalization. We say that, a + √b is the rationalizing factor of a - √b, similarly a - √b is the rationalizing factor of a + √b.
There is an irrational number suppose 5 + √7, let us multiply 5 - √7 and the product of these two irrational numbers would be a rational number. (5 + √7) . (5 - √7) = 5² - (√7)² = 25 – 7 = 18, this process is called rationalization. We say that, 5 + √7 is the rationalizing factor of 5 - √7, similarly 5 - √7 is the rationalizing factor of 5 + √7.
If, a + √b is an irrational number then a - √b is the rationalizing factor of a + √b, similarly a + √b is the rationalizing factor of a - √b.
So, a - √b and a + √b are said to be conjugate to each other.
7
Example. Rationalize the denominator of ---------- 5 + √7
7 7 5 - √7
-------- = --------- X ----------
5 + √7 5 + √7 5 - √7
7 (5 - √7)
= ------------------------
(5 + √7) (5 - √7)
35 - 7√7 35 - 7√7 35 - 7√7
= ------------ = ----------- = -----------
5² - (√7)² 25 – 7 18
35 7√7
= --------- - --------- (Ans.)
18 18