Question

Rationalise the denominator:

`(sqrt(6) - sqrt(5))/(sqrt(6) + sqrt(5))`

Answer 1

To rationalize the denominator of the expression

`(sqrt(6) - sqrt(5))/(sqrt(6) + sqrt(5))`,

we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is `sqrt(6) - sqrt(5)`.

`(sqrt(6) - sqrt(5))/(sqrt(6) + sqrt(5)) xx (sqrt(6) - sqrt(5))/(sqrt(6) - sqrt(5)) = ((sqrt(6) - sqrt(5))^2)/((sqrt(6))^2 - (sqrt(5))^2`

Step 1: Simplify the denominator

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

`(sqrt(6))^2 - (sqrt(5))^2`

= 6 – 5

= 1

Step 2: Expand the numerator

Now, expand `(sqrt(6) - sqrt(5))^2`:

`(sqrt(6) - sqrt(5))^2`

= `(sqrt(6))^2 - 2(sqrt(6))(sqrt(5)) + (sqrt(5))^2`

= `6 - 2sqrt(30) + 5`

= `11 - 2sqrt(30)`

Step 3: Put everything together

Thus, the expression becomes:

`(11 - 2sqrt(30))/1`

= `11 - 2sqrt(30)`