<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] First rewrite $5+2\sqrt{3} = (a-\sqrt{3b})(7+\sqrt{3})$.
Now suppose $b=3n^2$, then we get $5+2\sqrt{3} = (a-3|n|)(7+\sqrt{3})$, which can not be solved for rational $a$ and $n$.
Next, suppose $b=n^2$, then we get $5+2\sqrt{3} = (a-|n|\sqrt{3})(7+\sqrt{3}) = 7a-3|n|+(a-7|n|)\sqrt{3}$ and hence $5=7a-3|n|$ and $2=a-7|n|$. This gives $9=-46|n|$, which has no solutions either for rational $a$ and $n$.
Other $b$ won't work either, since you'll be left with a $\sqrt{b}$. Thus your equation has no rational solutions.
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