Question

Rationalise the denominator of the following and hence find their value by using `sqrt(2) = 1.414` and `sqrt(3) = 1.732`:

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

Answer 1

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

Step 1: Rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator `3sqrt(3) + 4sqrt(2)`:

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

= `(5(3sqrt(3) + 4sqrt(2)))/((3sqrt(3))^2 - (4sqrt(2))^2`

Step 2: Simplify the denominator using difference of squares:

`(3sqrt(3))^2`

= 9 × 3

= 27

`(4sqrt(2))^2`

= 16 × 2

= 32

So, 27 – 32 = –5.

Step 3: Substitute denominator and expand the numerator:

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

= `(15sqrt(3) + 20sqrt(2))/(-5)`

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

Step 4: Use the given approximations `sqrt(2) = 1.414, sqrt(3) = 1.732`:

–3 × 1.732 – 4 × 1.414

= –5.196 – 5.656

= –10.852