From a wikipedia entry section about the simplified form of a radical expression, only non-nested radical expression is covered. There are no hints about nested radical expressions. Some common sense may suggest that non-nested radical expression is simpler than nested radical expression. For example, $\sqrt{2}+\sqrt{6}$ should be simpler than $\sqrt{8 + 4 \sqrt{3}}$.
However, if it comes to different form with the same value, it does not seem trivial to determine which expression is simpler, especially when each looks tidier in different aspects. For an example of the value of $2\tan(48°)$, i.e. $2\tan(\frac{4\pi}{15})$:
$$3\sqrt{3}-\sqrt{15}-2\sqrt{10+2\sqrt{5}}+\sqrt{50+10\sqrt{5}}$$
and
$$\sqrt{50-22\sqrt{5}}+\sqrt{42-18\sqrt{5}}$$
yields the same value.
While it is true that $\sqrt{42-18\sqrt{5}}$ is actually $3\sqrt{3}-\sqrt{15}$ where the later looks simpler, I am uncertain whether $(\sqrt{5}-2)\sqrt{10+2\sqrt{5}}$ should be better written as
$$\sqrt{50-22\sqrt{5}}$$ (least radical symbol), $$\sqrt{50+10\sqrt{5}}-2\sqrt{10+2\sqrt{5}}$$ (expanded form), or even decomposing $\sqrt{10+2\sqrt{5}}$ to $\sqrt{5+2\sqrt{5}}+\sqrt{5-2\sqrt{5}}$ to make the numbers in the radicals "smaller".
The main question is, is there any existing definition of a unique simplest radical expression such that, let's say, a constructible number, can be represented in only one "simplest" way that two radical expressions are said to equal only if the "simplest" expression is identical? (Just like prime factorization in natural numbers) Or should a "minimal polynomial" be used instead for such purpose?