Question

Rationalise the denominator:

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

Answer 1

To rationalize the denominator of the expression

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

we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is `sqrt(5) + sqrt(3)`.

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

Now, simplify the denominator using the identity (a – b)(a + b) = a² – b²:

`(sqrt(5))^2 - (sqrt(3))^2`

= 5 – 3

= 2

Now, expand the numerator:

`(sqrt(5) + sqrt(3))^2`

= `(sqrt(5))^2 + 2(sqrt(5))(sqrt(3)) + (sqrt(3))^2`

= `5 + 2sqrt(15) + 3`

= `8 + 2sqrt(15)`

Thus, the expression becomes:

`(8 + 2sqrt(15))/2`

Now, simplify the result by dividing each term in the numerator by 2:

`4 + sqrt(15)`