Perfect power

A positive integer $n$ is a perfect power if there are integers $m, k$ such that $k \geq 2$ and $m^k = n$ . In particular, $n$ is said to be a perfect $k$ th power.

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

For example, $64 = 8^2 = 4^3 = 2^6$ , so 64 is a perfect 2nd, 3rd and 6th power.

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