If both `a` and `b` are rational numbers, find the values of `a` and `b` in each of the following equalities : `(sqrt(3)-1)/(sqrt(3)+1)=a+bsqrt(3)`

Explanation

To solve the equation \(\frac{\sqrt{3}-1}{\sqrt{3}+1} = a + b\sqrt{3}\), where \(a\) and \(b\) are rational numbers, we will rationalize the left-hand side of the equation.

### Step-by-Step Solution:

1. **Rationalize the Denominator**:
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3}-1\):
\[
\frac{\sqrt{3}-1}{\sqrt{3}+1} \cdot \frac{\sqrt{3}-1}{\sqrt{3}-1} = \frac{(\sqrt{3}-1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)}
...