You can decide whether a given element in a torsion-free hyperbolic group is a square.

Suppose $\Gamma=\langle a_1,\ldots,a_k\rangle$ and $a_0\in\Gamma$ is the element you're interested in. You can say $\Gamma=\langle a_0,a_1,\ldots,a_k\rangle$, so then the validity of following sentence can be decided by [[Sela, §4]][1]:
$$\exists x\ (a_0=x^2)$$
Technically this should begin with $\exists y$ to make an AE sentence, but the AE sentence 
$$\forall y\exists x\ (y=y)\wedge(a_0=x^2)$$
is equivalent. 

[1]: https://mathscinet.ams.org/mathscinet-getitem?mr=2520356