Rationalisation of Surds
Rationalising factor is a term with which a term is multiplied or divided to make the whole term rational.
(i) √3 is a rationalising factor of √3 (since √3 × √3 = the rational number 3)
(ii) 7√54 is a rationalising factor of 7√53 (since their product = 7√57 = 5 , a rational)
1. In the example (i) above, can √12 also be a rationalising factor? Can you think of any other number as a rationalising factor for √3 ?
2. Can you think of any other number as a rationalising factor for 7√53 in example (ii) ?
3. If there can be many rationalising factors for an expression containing a surd, is there any advantage in choosing the smallest among them for manipulation?
Identify a rationalising factor for each one of the following surds and verify the same in each case:
(i) √18 (ii) 5√12 (iii) 3√49 (iv) 1/√8
Can you guess a rationalising factor for 3 + √2 ? This surd has one rational part and one radical part. In such cases, the rationalising factor has an interesting form.
A rationalising factor for 3 + √2 is 3 − √2 . You can very easily check this.
(3+ √2)(3− √2) = 32− (√2)2
= 9 −2
= 7, a rational.
What could be the rationalising factor for a + √b where a and b are rational numbers? Is it a − √b ? Check it. What could be the rationalising factor for √a + √b where a and b are rational numbers? Is it √a − √b ? Or, is it − √a + √b ? Investigate.
Surds like a + √b and a − √b are called conjugate surds. What is the conjugate of √b +a ? It is − √b +a . You would have perhaps noted by now that a conjugate is usually obtained by changing the sign in front of the surd!
Rationalise the denominator of
(i) Multiply both numerator and denominator by the rationalising factor √14