Question

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

`a - bsqrt(3) = (4sqrt(3) - 7)/(4sqrt(3) + 7)`

Answer 1

Given: `a - bsqrt(3) = (4sqrt(3) - 7)/(4sqrt(3) + 7)`where a and b are rational numbers.

Stepwise calculation:

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

`(4sqrt(3) - 7)/(4sqrt(3) + 7) xx (4sqrt(3) - 7)/(4sqrt(3) - 7)`

= `(4sqrt(3) - 7)^2/((4sqrt(3))^2 - 7^2)`

2. Calculate numerator:

`(4sqrt(3) - 7)^2 = (4sqrt(3))^2 + (-7)^2 - 2 xx 4sqrt(3) xx 7`

= `16 xx 3 + 49 - 56sqrt(3)`

= `48 + 49 - 56sqrt(3)`

= `97-56sqrt(3)`

3. Calculate denominator:

`(4sqrt(3))^2 - 7^2`

= 16 × 3 – 49

= 48 – 49

= –1

4. So, `(4sqrt(3) - 7)^2/((4sqrt(3))^2 - 7^2)`

= `(97 - 56sqrt(3))/(-1)`

= `-97 + 56sqrt(3)`

5. Equate with the left-hand side expression:

`a - bsqrt(3) = -97 + 56sqrt(3)`

Comparing rational and irrational parts:

a = –97

–b = 56

⇒ b = –56