Question

Rationalise the denominator of the following:

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

Answer 1

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

Step 1: Identify the conjugate of the denominator `4 + sqrt(7)`, which is `4 - sqrt(7)`. 

Step 2: Multiply the numerator and denominator by the conjugate to rationalise the denominator:

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

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

Step 3: Simplify the denominator using the difference of squares formula:

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

= 16 – 7

= 9

Step 4: Expand the numerator:

`3(4 - sqrt(7)) = 12 - 3sqrt(7)`

Step 5: Substitute the simplified numerator and denominator:

`(12 - 3sqrt(7))/9`

Step 6: Simplify the fraction by dividing numerator terms by 9:

`12/9 - (3sqrt(7))/9 = 4/3 - sqrt(7)/3`