Question

Rationalise the denominator of `(2sqrt(5) - 3)/(3sqrt(2) - 4)`.

Answer 1

Let’s rationalise the expression:

`(2sqrt(5) - 3)/(3sqrt(2) - 4)`

Multiply numerator and denominator by the conjugate of the denominator:

`(2sqrt(5) - 3)/(3sqrt(2) - 4) xx (3sqrt(2) + 4)/(3sqrt(2) + 4)`

Step 1: Denominator

`(3sqrt(2) - 4)(3sqrt(2) + 4)`

= `(3sqrt(2))^2 - (4)^2`

= 9 × 2 – 16

= 18 – 16

= 2

Step 2: Numerator

`(2sqrt(5) - 3)(3sqrt(2) + 4)`

1. `2sqrt(5) xx 3sqrt(2)`

= `6sqrt(10)`

2. `2sqrt(5) xx 4`

= `8sqrt(5)`

3. `-3 xx 3sqrt(2)`

= `-9sqrt(2)`

4. –3 × 4 

= –12

Now combine:

`6sqrt(10) + 8sqrt(5) - 9sqrt(2) - 12`

So the rationalised expression is:

`(6sqrt(10) - 9sqrt(2) + 8sqrt(5) - 12)/2`