Question

Rationalise the denominator of `(4 - sqrt(5))/(3 + sqrt(5))`.

Answer 1

To rationalise the denominator of `(4 - sqrt(5))/(3 + sqrt(5))`

Multiply numerator and denominator by the conjugate of the denominator `(3 - sqrt(5))`:

`(4 - sqrt(5))/(3 + sqrt(5)) xx (3 - sqrt(5))/(3 - sqrt(5)) = ((4 - sqrt(5))(3 - sqrt(5)))/((3 + sqrt(5))(3 - sqrt(5))`

Step 1: Denominator

`(3 + sqrt(5))(3 - sqrt(5))`

= 9 – 5

= 4

Step 2: Numerator

`(4 - sqrt(5))(3 - sqrt(5))`

= `4 xx 3 - 4sqrt(5) - 3sqrt(5) + (sqrt(5))^2`

= `12 - 4sqrt(5) - 3sqrt(5) + 5`

= `(17 - 7sqrt(5))/4`