What would be the value of $a$ and $b$ in following rational expression? If $(5 + 2\sqrt{3})/(7 + \sqrt{3}) = (a - \sqrt{3b})$,
How do I find the value of $a$ and $b$ where $a$ and $b$ are rational numbers?

What was written was
$$\frac{5 + 2\sqrt{3}}{7 + \sqrt{3}} = a - \sqrt{3b},$$
but I'm going to boldly assume that's a typo and what was really intended was
$$\frac{5 + 2\sqrt{3}}{7 + \sqrt{3}} = a - \sqrt{3}b.$$
Multiply the numerator and denominator by the conjugate of $7+\sqrt{3}$, which is $7-\sqrt{3}$:
\begin{align}
& \frac{5 + 2\sqrt{3}}{7 + \sqrt{3}} = \frac{(5 + 2\sqrt{3})(7 - \sqrt{3})}{(7 + \sqrt{3})(7 - \sqrt{3})} = \frac{29 + 9\sqrt{3}}{7^2-3} = \frac{29 + 9\sqrt{3}}{46} \\[10pt]
& = \frac{29}{46} + \frac{9}{46}\sqrt{3} = \frac{29}{46} - \frac{-9}{46}\sqrt{3}.
\end{align}
So $a=\dfrac{29}{46}$ and $b=\dfrac{-9}{46}$.
