There is no standard form for such expressions. However, some forms may prove more convenient for certain applications, e.g. transforming the radicands to be squarefree helps to ascertain multiplicative independence, hence linear independence (cf. this theorem). But even that is not necessary since it can also be tested without squarefree normalization by instead verifying that no subset of the radicals has rational product.
The fact that there is no computationally effective normal form for such expressions is a reflection of the innate complexity of their arithmetic. The complexity of even simpler problems involving
radicals is currently unknown. For example, no polynomial time
algorithm is known for determining the sign of a sum of real
radicals $\,\sum c_i q_i^{1/r_i},\,$ where $\,c_i, q_i\,$ are rationals and
$\,r_i\,$ is a positive integer. Such sums play an important role in
various geometric problems (e.g. Euclidean shortest paths and
traveling salesman tours). Testing whether such a sum
of radicals is zero can be decided in polynomial time, but this is
of no help in determining the sign (it only shows that if sign
testing is in NP then it is already in NP $\cap $ co-NP).
Generally the context will determine what type of normal or standard forms prove most convenient. For example, denesting of radicals is governed by a structure theorem that employs a certain normalized form, e.g.
$$ \sqrt{\sqrt[3]5 - \sqrt[3]4} \;\;=\; \frac{1}3 (\sqrt[3]2 + \sqrt[3]{20} - \sqrt[3]{25})$$
normalises to $$ \sqrt{18\ (\sqrt[3]10 - 2)} \;\;=\; 2 + 2\ \sqrt[3]{10} - \sqrt[3]{10}^2 $$
and
$$ \sqrt{12 + 5\ \sqrt 6} \;\;=\; (\sqrt 2 + \sqrt 3)\ 6^{1/4} $$
normalises to
$$ \sqrt{\frac{1}3 \sqrt{6}\: (12 + 5\ \sqrt 6)} \;\;=\; 2 + \sqrt{6} $$
See here for more on this Denesting Structure Theorem.