Question

Find the value of a and b in the following:

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

Answer 1

We are asked to find a and b such that:

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

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

The conjugate of `7 + 4sqrt(3)` is `7 - 4sqrt(3)`.

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

Step 2: Expand numerator

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

= `14 - 8sqrt(3) + 7sqrt(3) - 12`

= `(14 - 12) + (-8sqrt(3) + 7sqrt(3))`

= `2 - sqrt(3)`

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

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

= 49 – 16 × 3

= 49 – 48

= 1

Step 4: Final simplification

`(2 - sqrt(3))/1 = 2 - sqrt(3)`

Step 5: Compare with `a + bsqrt(3)`

a = 2, b = –1