Why is the following language undecidable?
This is a direct consequence of Rice's theorem. A word $w$ is in $L$ if $T(M_w)$ satisfies the following semantic property:

$T(M_w)$ is non-empty, and for any $x \in T(M_w)$, we have $x^2 \in T(M_w)$ and $x^3 \notin T(M_w)$.

In order to show that this language is undecidable according to Rice's theorem, we need to exhibit two Turing machines $w_1$ and $w_2$: one which doesn't satisfy the property, and another one which does satisfy it. We can take $w_1$ to be some machine such that $T(M_{w_1}) = \emptyset$, and $w_2$ to be some machine such that $T(M_{w_2}) = \{ a^{2^n} : n \geq 0 \}$.