Question

Rationalise the denominator:

`(sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2))`

Answer 1

To rationalize the denominator of the expression

`(sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2))`,

we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is `sqrt(5) + sqrt(2)`.

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

Step 1: Simplify the denominator

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

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

= 5 – 2

= 3

Step 2: Expand the numerator

Now, expand `(sqrt(5) + sqrt(2))^2`:

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

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

= `5 + 2sqrt(10) + 2`

= `7 + 2sqrt(10)`

Step 3: Put everything together

Thus, the expression becomes:

`(7 + 2sqrt(10))/3`