Aus Online Mathematik Brückenkurs 1

One way to compare the two numbers is to rewrite the power \displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} so that it has the same exponent as \displaystyle 400^{\frac{1}{3}},

\displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} = 5^{\frac{1}{3}\cdot 4} = 5^{4\cdot\frac{1}{3}} = \bigl(5^{4}\bigr)^{\frac{1}{3}} = \bigl(5\cdot 5\cdot 5\cdot 5\bigr)^{\frac{1}{3}} = 625^{\frac{1}{3}}\,.

Now, we see that \displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} > 400^{\frac{1}{3}}, because \displaystyle 625 > 400 and the exponent 1/3 is positive.