Perfect power

A positive integer $n$ is a perfect power if there exist integers $m, k$ such that $k \geq 2$ and $n = m^k$ . In particular, $n$ is said to be a perfect $k$ th power. For example, $64 = 8^2 = 4^3 = 2^6$ , so 64 is a perfect 2nd, 3rd and 6th power.

We restrict $k \geq 2$ only because "being a perfect $1$ st power" is a meaningless property: every integer is a $1$ st power of itself.

Perfect second powers are usually known as perfect squares and perfect third powers are usually known as perfect cubes. This is because the area of a square (volume of a cube) with integer edge is equal to the second (respectively third) power of that edge, and so is a perfect second (respectively third) power.

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Perfect power