Squares, cubes and roots

A perfect square has even prime exponents because each factor is repeated twice. A perfect cube has prime exponents divisible by \(3\). Taking the corresponding root divides those exponents by \(2\) or \(3\).

Example

\(144=2^4\times3^2\), so \(\sqrt{144}=2^2\times3=12\). Also \(216=2^3\times3^3\), so \(\sqrt[3]{216}=2\times3=6\).

For \(72=2^3\times3^2\), the least positive integer multiplier making a square is \(2\): \(72\times2=144\). To make a cube, complete both exponents to \(3\): multiply by \(3\), giving \(216\).

For the least positive integer divisor making \(72\) a square, remove the unmatched factor \(2\): \(72\div2=36\). The symbol \(\sqrt{36}\) means \(6\); solving \(x^2=36\) instead gives \(x=6\) or \(x=-6\).

Recall cubes: \(1^3=1,2^3=8,3^3=27,4^3=64,5^3=125,10^3=1000\), and their corresponding roots.