I've just recently learned about denesting a radical and noted that $\sqrt{a\pm b\sqrt{c}}=\sqrt{\frac{a+\sqrt{a^2-b^2c}}2}+\sqrt{\frac{a-\sqrt{a^2-b^2c}}2}$, which doesn't seem to be able to hold if $b^2c>a^2$. It also doesn't seem to hold for complex $a,b,c$.
This also appears to have the affect that $\sqrt{a+b\sqrt{c}}=\sqrt{a-b\sqrt{c}}$, creating an obvious problem.