Simplification of expressions containing radicals As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$.
But we can also find the roots using Cardano's method, which leads to
$$x = \sqrt[3]{\sqrt{28/27} + 1} - \sqrt[3]{\sqrt{28/27} - 1}$$
and two other roots.
It's easy to check numerically that this expression is really equal to $1$, but is there a way to derive it algebraically which isn't equivalent to showing that this expression satisfies $f(x) = 0$?

There are very general algorithms known for radical denesting. Below is the structure theorem which lies at the foundation of these algorithms. It widely generalizes the heuristic employed by Qiaochu in his answer. It may be employed heuristically - in a similar manner as Qiaochu - to perform complicated denestings, without requiring much comprehension of the underlying theory.
In Bloemer's papers FOCS '91 &  FOCS '92 & Algorithmica 2000  you will find polynomial-time algorithms for radical denesting. Informally, the key  Denesting Structure Theorem says that if a radical $\rm\, r^{1/d} \,$ denests in any radical extension $\rm\, F' \,$ of its base field $\rm\, F \,$, then a suitable multiple $\rm\, q b\:\!\:\! r \,$ of
the radicand $\rm\:\! r\:\! $ must already denest in the field $\rm\, F' \,$ defined by the
radicand. More precisely
Denesting Structure Theorem$\,\, \,$  Let $\rm\, F \,$ be a real field and
$\rm\, F' = F(q_1^{1/d1},\ldots,q_k^{1/dk}) \,$ be a real radical extension of $\rm\, F \,$ of degree $\rm\, n \,$. Let $\rm\, B = \{b_0,\ldots, b_{n-1}\}$ be the standard
basis of $\rm\, F' \,$ over $\rm\, F \,$. If $\rm\, r \,$ is in $\rm\, F' \,$ and $\rm\, d \,$ is a positive integer such
that $\rm\, r^{1/d} \,$ denests over $\rm\, F \,$ using only real radicals, that is,
$\rm\, r^{1/d} \,$ is in $\rm\, F(a_1^{1/t_1},\ldots,a_m^{1/t_m}) \,$ for some positive integers
$\rm\, t_i \,$ and positive $\rm\, a_i \in F \,$, then there exists a nonzero $\rm\, q \in F \,$ and a
$\rm\, b \in B \,$ such that $\rm\, (q b r)^{1/d}\! \in F' \,$.
I.e. multiplying the radicand by a $\rm\, q \,$ in the base field $\rm\, F \,$
and a power product $\rm\, b = q_1^{e_1/d_1}\cdots q_k^{e_k/d_k} \,$ we can
normalize any denesting so that it denests in the field defined
by the radicand. E.g.
$$ \sqrt{\sqrt[3]5 - \sqrt[3]4} \,\,=\, \frac{1}3 (\sqrt[3]2 + \sqrt[3]{20} - \sqrt[3]{25})$$
normalises to $$\qquad \sqrt{18\ (\sqrt[3]10 - 2)} \,\,=\, 2 + 2\ \sqrt[3]{10} - \sqrt[3]{10}^2\,\in\,\Bbb Q(\sqrt[3]{10}) $$
An example with nontrivial $\rm\,b$
$$  \sqrt{12 + 5\ \sqrt 6} \,\,=\, (\sqrt 2 + \sqrt 3)\ 6^{1/4}\qquad\quad $$
normalises to
$$ \sqrt{\frac{1}3 \sqrt{6}\, (12 + 5\ \sqrt 6)} \,\,=\, 2 + \sqrt{6}\,\in\, \Bbb Q(\sqrt 6)\qquad\qquad\ \ \  $$
Here $\rm\, F=\mathbb Q,\ F' = \mathbb Q(\sqrt 6),\ n=2,\ B = \{1,\sqrt 6\},\ d=2,\ q=1/3,\ b= \sqrt 6\,$.
The structure theorem also hold for complex fields except that
in this case one has to assume that $\rm\, F \,$ contains enough roots of
unity (which may be computationally expensive in practice, to
wit doubly-exponential complexity).
Note that 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 $\rm\, \sum{c_i\, q_i^{1/r_i}} \,$ where $\rm\, c_i,\, q_i \,$ are rational numbers and
$\rm\, r_i \,$ is a positive integer. Such sums play an important role in
various geometric problems (e.g. Euclidean shortest paths and
traveling salesman tours). Even though testing whether such a sum
of radicals is zero can be decided in polynomial time, this is
of no help in determining the sign, it only shows that if sign
testing is in $\rm\, NP \,$ then it is already in $\rm\, NP \cap \text{co-NP} \,$.
