Question

Rationalise the denominator of `(2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2))`.

Answer 1

To rationalise the expression:

`(2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2))`

Step 1: Multiply numerator and denominator by the conjugate of the denominator:

`(2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2)) xx (2sqrt(3) - sqrt(2))/(2sqrt(3) - sqrt(2))`

= `(2sqrt(3) - sqrt(2))^2/((2sqrt(3) + sqrt(2))(2sqrt(3) - sqrt(2))`

Step 2: Denominator

`(2sqrt(3))^2 - (sqrt(2))^2`

= 4 × 3 – 2

= 12 – 2

= 10

Step 3: Numerator

`(2sqrt(3) - sqrt(2))^2 = (2sqrt(3))^2 - 2 xx 2sqrt(3) xx sqrt(2) + (sqrt(2))^2`

• `(2sqrt(3))^2 = 4 xx 3 = 12`
• `2 xx 2sqrt(3) xx sqrt(2) = 4sqrt(6)`
• `(sqrt(2))^2 = 2`

So numerator `12 - 4sqrt(6) + 2 = 14 - 4sqrt(6)`

` (14 - 4sqrt(6))/10 = (7 - 2sqrt(6))/5`