Question

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

`a + bsqrt(2) = (4 + 3sqrt(2))/(4 - 3sqrt(2))`

Answer 1

Given: `a + bsqrt(2) = (4 + 3sqrt(2))/(4 - 3sqrt(2))`

Step-wise calculation:

1. Rationalise the denominator on the right-hand side:

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

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

2. Calculate numerator:

`(4 + 3sqrt(2))^2`

= `4^2 + 2 xx 4 xx 3sqrt(2) + (3sqrt(2))^2`

= `16 + 24sqrt(2) + 9 xx 2`

= `16 + 24sqrt(2) + 18`

= `34 + 24sqrt(2)`

3. Calculate denominator:

`4^2 - (3sqrt(2))^2`

= 16 – 9 × 2

= 16 – 18

= –2

4. Therefore, `(4 + 3sqrt(2))/(4 - 3sqrt(2))`

= `(34 + 24sqrt(2))/(-2)`

= `-17 - 12sqrt(2)`

From the equality, `a + bsqrt(2) = -17 - 12sqrt(2)`.

Comparing rational and irrational parts: a = –17, b = –12.