Roots of Perfect Squares and Cubes

Property The square root of a perfect square is an integer. The cube root of a perfect cube is an integer. If $x = n^2$ for some integer $n$, then $\sqrt{x} = n$. If $y = m^3$ for some integer $m$, then $\sqrt[3]{y} = m$.

Examples The square root of the perfect square $81$ is an integer: $\sqrt{81} = \sqrt{9^2} = 9$. The cube root of the perfect cube $64$ is an integer: $\sqrt[3]{64} = \sqrt[3]{4^3} = 4$. The cube root of the negative perfect cube $ 125$ is an integer: $\sqrt[3]{ 125} = \sqrt[3]{( 5)^3} = 5$.

Explanation A perfect square is the result of squaring an integer, and a perfect cube is the result of cubing an integer. When you take the square root of a perfect square, you are simply finding the original integer that was squared. Similarly, taking the cube root of a perfect cube reveals the original integer that was cubed. This is why the roots of perfect squares and perfect cubes are always integers.