Certain expressions of the form specified in the question can be simplified. $\sqrt{7+4\sqrt{3}}=2+\sqrt{3}$, for example.
My attempt:
Assumptions: a and b are rational.
Assume that $\sqrt{x+y\sqrt{n}}=a+b\sqrt{n}$. Squaring both sides results in $x+y\sqrt{n}=a^2+nb^2+2ab\sqrt{n}$. Because a and b are rational (assumption), $a^2+nb^2=x$ and $2ab=y\implies{b=\frac{y}{2a}}$. Plugging this result into the first equation, we get $a^2+\frac{ny^2}{a^2}=x\implies(a^2)^2-xa^2+ny^2=0$. Using the quadratic formula, $a^2=\frac{-x\pm\sqrt{x^2-4ny^2}}{2}\implies{a=\sqrt{\frac{-x\pm\sqrt{x^2-4ny^2}}{2}}}$. But this itself has nested square roots!
My question:
Is it always possible to simplify $\sqrt{x+y\sqrt{n}}$ in the form of $a+b\sqrt{n}$ without $a$ or $b$ containing any nested roots themselves? If so, how is it possible? If not, when is it possible?
