Question

Rationalise the denominator of the following:

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

Answer 1

Given expression: `(3 + sqrt(2))/(3 - sqrt(2))`

Step 1: Identify the rationalising factor of the denominator.

The denominator is `3 - sqrt(2)`.

The rationalising factor will be `3 + sqrt(2)` conjugate.

Step 2: Multiply the numerator and denominator by the conjugate of the denominator:

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

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

Step 3: Simplify the denominator using difference of squares: 9 – 2 = 7.

Step 4: Expand numerator:

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

= `3^2 + 2 xx 3 xx sqrt(2) + (sqrt(2))^2`

= `9 + 6sqrt(2) + 2`

= `11 + 6sqrt(2)`

Step 5: Write simplified form: `(11 + 6sqrt(2))/(7)`.