<s>[INST] 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?
 [/INST] $$\frac{5+2\sqrt{3}}{7+\sqrt{3}} = a - \sqrt{3b}$$
$$5+2\sqrt{3}=(a-\sqrt{3b})(7+\sqrt{3})$$
$(a-\sqrt{3b})(7+\sqrt{3})=7a+a\sqrt{3}-7\sqrt{3b}-3\sqrt{b}$
Now we have two equotation for two unknown $a$ and $b$:
$5 = 7a-3\sqrt{b}$
and$2\sqrt{3}=(a-\sqrt{b})\sqrt{3} \implies 2=a-\sqrt{b}$.
Now
$\sqrt{b}=\frac{7a-5}{3}=a-2$
$7a-5=3a-6 \implies a = -\frac{1}{4}$
$\sqrt{b} = a - 2 = -\frac{1}{4} - \frac{8}{4}=-\frac{9}{4} \implies b = \frac{81}{16}$
So, 
$$a = -\frac{1}{4}, b = \frac{81}{16}$$
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