Question

Rationalise the denominator:

`9/(sqrt(11) + sqrt(5))`

Answer 1

We are given:

`9/(sqrt(11) + sqrt(5))`

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

`9/(sqrt(11) + sqrt(5)) xx (sqrt(11) - sqrt(5))/(sqrt(11) - sqrt(5))`

= `(9(sqrt(11) - sqrt(5)))/((sqrt(11) + sqrt(5))(sqrt(11) - sqrt(5))`

Step 2: Use identity (a + b)(a – b) = a² – b²:

`(sqrt(11) + sqrt(5))(sqrt(11) - sqrt(5))`

= `(sqrt(11))^2 - (sqrt(5))^2`

= 11 – 5

= 6

Step 3: Final simplification:

`(9(sqrt(11) - sqrt(5)))/6`

= `3/2 (sqrt(11) - sqrt(5))`