Lessons from Power Function

Three things to remember:

$1.$ Squaring reduces the problem size by half, giving you $O(\log n)$ instead of $O(n)$

$2.$ Odd exponents need one extra multiplication by the base

$3.$ Negative exponents flip to reciprocals using the rule $x^{-n} = 1/x^n$

This pattern appears in modular exponentiation, matrix exponentiation, and other divide-and-conquer algorithms. Halving the work repeatedly is a powerful trick.