Question

Rationalise the denominator of the following:

`(7 - 4sqrt(3))/(7 + 4sqrt(3))`

Answer 1

Given expression: `(7 - 4sqrt(3))/(7 + 4sqrt(3))`

Step 1: To rationalise the denominator, multiply the numerator and denominator by the conjugate of the denominator `7 - 4sqrt(3)`:

`(7 - 4sqrt(3))/(7 + 4sqrt(3)) xx (7 - 4sqrt(3))/(7 - 4sqrt(3))`

= `(7 - 4sqrt(3))^2/((7)^2 - (4sqrt(3))^2`

Step 2: Calculate the denominator:

`7^2 - (4sqrt(3))^2`

= 49 – 16 × 3

= 49 – 48

= 1

Step 3: Expand the numerator: 

`(7 - 4sqrt(3))^2`

= `7^2 - 2 xx 7 xx 4sqrt(3) + (4sqrt(3))^2`

= `49 - 56sqrt(3) + 16 xx 3`

= `49 - 56sqrt(3) + 48`

= `97 - 56sqrt(3)`

Step 4: So the expression becomes:

`(97 - 56sqrt(3))/1 = 97 - 56sqrt(3)`