Question

Find the value of a and b in the following:

`sqrt(3)/(3sqrt(2) + 2sqrt(3)) = a + bsqrt(6)`

Answer 1

We are given:

`sqrt(3)/(3sqrt(2) + 2sqrt(3)) = a + bsqrt(6)`

We are to express this in the form `a + bsqrt(6)`.

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

The conjugate of `3sqrt(2) + 2sqrt(3)` is `3sqrt(2) - 2sqrt(3)`

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

Step 2: Expand the numerator:

`sqrt(3)(3sqrt(2) - 2sqrt(3))`

= `3sqrt(6) - 2 xx 3`

= `3sqrt(6) - 2sqrt(9)`

= `3sqrt(6) - 6`

Step 3: Expand the denominator using identity (a + b)(a – b) = a² – b²:

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

= 9 × 2 – 4 × 3

= 18 – 12

= 6

Step 4: Final simplification:

`(3sqrt(6) - 6)/6`

= `(3sqrt(6))/6 - 6/6`

= `1/2 sqrt(6) - 1`

Rewriting:

`-1  + 1/2 sqrt(6)`

`a = -1, b = 1/2`