Surds Can Be Multiplied

In a nutshell: For all numbers \(a\) and \(b\), and integer \(n\), \[ \sqrt[n]{a\vphantom{b}\,} \sqrt[n]{\,\vphantom{a}b\,} = \sqrt[n]{ab}\,. \]

Explain, please

Well, \begin{align} \left(\sqrt[n]{a\,}\right)^n &= a, &&\text{and}\\ \left(\sqrt[n]{\,b\,}\right)^n &= b, &&\text{by definition, so}\\ \left(\sqrt[n]{a\vphantom{b}\,} \sqrt[n]{\,\vphantom{a}b\,}\right)^n&=ab\\ \sqrt[n]{a\vphantom{b}\,} \sqrt[n]{\,\vphantom{a}b\,} &= \sqrt[n]{ab}\,. \end{align}

Explain more please

[Sigh.] Okay.

Watch this precious space guys!