Question

Rationalise the denominator of `(2sqrt(5) - sqrt(10))/(2sqrt(5) + sqrt(10))`.

Answer 1

To rationalise the expression:

`(2sqrt(5) - sqrt(10))/(2sqrt(5) + sqrt(10))`

Step 1: Multiply numerator and denominator by the conjugate of the denominator:

`(2sqrt(5) - sqrt(10))/(2sqrt(5) + sqrt(10)) xx (2sqrt(5) - sqrt(10))/(2sqrt(5) - sqrt(10))`

= `(2sqrt(5) - sqrt(10))^2/((2sqrt(5) + sqrt(10))(2sqrt(5) - sqrt(10))`

Step 2: Denominator

Use the identity `(a + b)(a - b) = a^2 - b^2`:

`(2sqrt(5))^2 - (sqrt(10))^2`

= 4 × 5 – 10

= 20 – 10

= 10

Step 3: Numerator

`(2sqrt(5) - sqrt(10))^2 = (2sqrt(5))^2 - 2 xx 2sqrt(5) xx sqrt(10) + (sqrt(10))^2`

`(2sqrt(5))^2 = 4 xx 5 = 20`

`2 xx 2sqrt(5) xx sqrt(10) = 4sqrt(50)`

= `4 xx sqrt(25 xx 2)`

= `4 xx 5sqrt(2)`

= `20sqrt(2)`

`(sqrt(10))^2 = 10`

So numerator becomes:

`20 - 20sqrt(2) + 10 = 30 - 20sqrt(2)`

⇒ `(30 - 20sqrt(2))/10`

= `30/10`

= `(20sqrt2)/10`

= `3 - 2sqrt(2)`