a) The example that it isn't necessary: $a = c = 2$, $b = d = 4$.

b) It shows that if a positive integer number $n$ has its square $n^2$ as a perfect cube of an integer number, then $n$ itself is a perfect cube of some integer number. Really, consider a factorization of $n$ into prime numbers $n = p_1^{m_1} \ldots p_k^{m_k}$, its square is $n^2 = p_1^{2m_1} \ldots p_k^{2m_k}$, if $p_1^{2m_1} \ldots p_k^{2m_k}$ is a perfect cube then for each $i = 1, k$ we have to hold the condition $2m_i \equiv 0 \pmod{3}$ which is equal to the condition $m_i \equiv 0 \pmod{3}$. Hence, the number $n$ is a perfect cube.

As $\frac{ab \cdot cd \cdot ea}{bc \cdot de} = a^2 = a_1^3$, then each of the numbers $a$, $b$, $c$, $d$, $e$ is a perfect cube.