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)` (ii) `(3+sqrt(7))/(3-sqrt(7))=a+bsqrt(7)` `(5+2sqrt(3))/(7+4sqrt(3))=a+bsqrt(3)` (iv) `(5+sqrt(3))/(7-4sqrt(3))=47a+sqrt(3)b` `(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))=a+bsqrt(15)` (iv) `(sqrt(2)+sqrt(3))/(3sqrt(2)-2sqrt(3))=1-bsqrt(3)`

Explanation

To solve the given equalities step by step, we will rationalize the left-hand side (LHS) of each equality and compare it with the right-hand side (RHS) to find the values of \( a \) and \( b \).

### (i) \(\frac{\sqrt{3}-1}{\sqrt{3}+1} = a + b\sqrt{3}\)

**Step 1: Rationalize the LHS.**
Multiply the numerator and denominator by the conjugate of the denominator:
\[
\frac{(\sqrt{3}-1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{(\sqrt{3}-1)^2}{(\sqrt{3})^2 - (1)^2}
...