How is this problem related to the study of algorithms and big O notation?
You need to be more patient. This problem is hinting at the repeated squaring algorithm for powering. The more general question is: How long does it take to compute $x^n$? Naively, you might think that it takes $n$ multiplications, but the repeated squaring algorithm can do it using $O(\log n)$ multiplications.

Repeated squaring is very important for cryptographic applications. Many cryptographic algorithms rely on computing modular powers: $x^p \pmod{n}$, where in general $p$ and $n$ could be very large (say roughly $2^{1024}$). It makes a huge difference whether you do it in $O(p)$ or in $O(\log p)$ time.