<s>[INST] How do I solve this square root problem? I need to solve the following problem:
$$\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=\,?$$
 [/INST] $\frac{7+\sqrt{5}}{7-\sqrt{5}}=\frac{7+\sqrt{5}}{7-\sqrt{5}}\cdot\frac{7+\sqrt{5}}{7+\sqrt{5}}=\frac{(7+\sqrt{5})^{2}}{49-5}=\frac{(7+\sqrt{5})^{2}}{44}$
Then you are taking the root of the quotient (note for $a,b >0$ $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$) and you get
$\frac{7+\sqrt{5}}{\sqrt{44}}=\frac{7+\sqrt{5}}{2\sqrt{11}}=\frac{\sqrt{11}(7+\sqrt{5})}{22}$
None of the answers proposed is correct: we can use the squared value we have calculated
$\frac{(7+\sqrt{5})^{2}}{44}=\frac{3}{11}+\frac{\sqrt{35}}{22}$
As you can see it is not rational, so you exclude $1$ and $2$
Then $(6 \pm \sqrt{35})^2= 36+35 \pm 12 \sqrt{35}$ and you can see that both of them are incorrect.
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