Question

Rationalise the denominator of `1/(sqrt(2) + sqrt(3) - sqrt(5))`.

Answer 1

Given the expression:

`1/(sqrt(2) + sqrt(3) - sqrt(5))`

Step 1: Multiply numerator and denominator by the conjugate expression to eliminate the square roots in the denominator.

Since there are three terms in the denominator, use the conjugate involving the terms changed accordingly.

One approach is to first consider the conjugate as:

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

Multiply numerator and denominator by this:

`1/(sqrt(2) + sqrt(3) - sqrt(5)) xx (sqrt(2) + sqrt(3) + sqrt(5))/(sqrt(2) + sqrt(3) + sqrt(5))`

This gives:

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

Step 2: Calculate the denominator:

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

= `(2 + 3 + 2sqrt(6)) - 5`

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

= `2sqrt(6)`

Step 3: The expression simplifies to:

`(sqrt(2) + sqrt(3) + sqrt(5))/(2sqrt(6))`

Step 4: To rationalise further, multiply numerator and denominator by `sqrt(6)`:

`((sqrt(2) + sqrt(3) + sqrt(5))sqrt(6))/(2sqrt(6)sqrt(6))`

= `(sqrt(12) + sqrt(18) + sqrt(30))/(2 xx 6)`

Step 5: Simplify the square roots in numerator:

`sqrt(12) = 2sqrt(3)`,

`sqrt(18) = 3sqrt(2)`,

`sqrt(30) = sqrt(30)`

So numerator becomes:

`2sqrt(3) + 3sqrt(2) + sqrt(30)`

Denominator is 12.

`1/(sqrt(2) + sqrt(3) - sqrt(5)) = (2sqrt(3) + 3sqrt(2) + sqrt(30))/12`