Rationalisation of Surds

Rationalisation of Surds

Rationalising factor is a term with which a term is multiplied or divided to make the whole term rational.

Examples:

(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)

Thinking Corner

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?

Progress Check

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

1. Conjugate Surds

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!

Example 2.27

Rationalise the denominator of 

Solution

(i) Multiply both numerator and denominator by the rationalising factor √14