Question

Rationalise the denominator of `(2 + sqrt(3))/(4 - sqrt(3))`.

Answer 1

To rationalise the denominator of `(2 + sqrt(3))/(4 - sqrt(3))`

Multiply numerator and denominator by the conjugate of the denominator `(4 + sqrt(3))`:

`(2 + sqrt(3))/(4 - sqrt(3)) xx (4 + sqrt(3))/(4 + sqrt(3)) = ((2 + sqrt(3))(4 + sqrt(3)))/((4 - sqrt(3))(4 + sqrt(3))`

Step 1: Denominator

`(4 - sqrt(3))(4 + sqrt(3))`

= `4^2 - (sqrt(3))^2`

= 16 – 3

= 13

Step 2: Numerator

Use distributive property:

`(2 + sqrt(3))(4 + sqrt(3))`

= `2 xx 4 + 2 xx sqrt(3) + sqrt(3) xx 4 + sqrt(3) xx sqrt(3)`

= `8 + 2sqrt(3) + 4sqrt(3) + 3`

= `(11 + 6sqrt(3))/13`