Question

If a and b are rational numbers, find the value of a and b:

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

Answer 1

Given: `a + bsqrt(6) = (2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2))` where a and b are rational numbers.

Stepwise calculation:

1. Rationalise the denominator:

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

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

2. Calculate numerator:

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

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

= `4 xx 3 - 4sqrt(6) + 2`

= `12 - 4sqrt(6) + 2`

= `14 - 4sqrt(6)`

3. Calculate denominator:

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

= 4 × 3 – 2

= 12 – 2

= 10

4. So the expression becomes:

`(14 - 4sqrt(6))/10`

= `14/10 - 4/10sqrt(6)`

= `7/5 - 2/5sqrt(6)`

5. Comparing with `a + bsqrt(6)`: 

`a = 7/5, b = -2/5`