Rationalisation : When surds occur in the denominator of a fraction, it is customary to rid the denominator of the radicals. The surd in the denominator is multiplied by an appropriate expression, such that the product is a rational number. The given surd and the expression by which it is multiplied are called rationalising factors of each other.
For example,
(i) \(a^{1-\frac{1}{n}}\) is the rationalising factor of \(a^\frac{1}{n}\),as \(a^\frac{1}{n}\) \(a^{1-\frac{1}{n}}\) = \(a^{\frac{1}{n}+1-\frac{1}{n}}\) = a1 = a, which is a rational number. Hence, the rationalising factor of \(6^\frac{1}{5}\) is \(6^{1-\frac{1}{5}}\) = \(6^\frac{1}{5}\).
(ii) (a + √b) is the rationalising factor of (a- √b) , as, (a + √b)(a - √b) = a2 – b, which is a rational. Hence, (3 + √2) is the rationalising factor of (3 - √2) .
(3 + √2) (3 - √2) = 9 – 2 = 7
(iii) (√a + √b) is the rationalising factor of (√a + √b) as (√a + √b)(√a - √b) = (√a)2 - (√b)2 = a - b, which is a rational number.
Note: Such binominal surds as (√a + √b) and (√a - √b)which differ only in the sign connecting their terms are said to be conjugate surds.
The product of conjugate surds is always a rational.
Thus, the process of multiplication of a surd by its rationalising factor is called rationalisation.
