Aus Online Mathematik Brückenkurs 1

Both \displaystyle 125 and \displaystyle 625 can be written as powers of \displaystyle 5,

\displaystyle \begin{align} & 125=5\centerdot 5=5\centerdot 5\centerdot 5=5^{3} \\ & \\ & 625=5\centerdot 125=5\centerdot 5^{3}=5^{4} \\ & \\ \end{align}

and this means that

\displaystyle \begin{align} & 125^{\frac{1}{2}}=\left( 5^{3} \right)^{\frac{1}{2}}=5^{3\centerdot \frac{1}{2}}=5^{\frac{3}{2}} \\ & \\ & 625=\left( 5^{4} \right)^{\frac{1}{3}}=5^{4\centerdot \frac{1}{3}}=5^{\frac{4}{3}} \\ \end{align}

From this, we see that \displaystyle 125^{\frac{1}{2}}>625^{\frac{1}{3}}, since the exponent \displaystyle {3}/{2}\; is bigger than \displaystyle {4}/{3}\; and the base \displaystyle 5 is bigger than \displaystyle 1.