I tried to use the cubic formula before, but was always stuck at simplifying cube roots. I had learned that you could always simply $\sqrt{a+\sqrt{b}}$.
However, can I always simply $\sqrt[3]{a+\sqrt{b}}$ as well? Or, is there a restriction to do so?
Edit:
By simplifying $\sqrt{a+\sqrt{b}}$, I mean denesting $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}$, which is the same as solving $m+n=a$, and $mn={b\over4}$. It may result in complex numbers.
Second Edit:
An example for denesting $\sqrt{1+\sqrt{2}}$, where $m+n=1$ and $mn={1\over2}$. Squaring the first equation then subtracting the second equation 4 times, m-n=$i$. As a result
$$\sqrt{1+\sqrt{2}}=\sqrt{1+i\over2}+\sqrt{1-i\over2}$$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $\sqrt{3-2\sqrt2}$ we can do it, but for $\sqrt{\sqrt2-1}$ it's impossible.
For $\sqrt[3]{5\sqrt2-7}$ we can do it, but for $\sqrt[3]{\sqrt2-1}$ it's impossible.
For $\sqrt[3]{\sqrt[3]2-1}$ we can do it, but for $\sqrt[3]{\sqrt[3]2+1}$ it's impossible.