Aus Online Mathematik Brückenkurs 1

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

\displaystyle \left( 5^{\frac{1}{3}} \right)^{4}=5^{\frac{1}{3}\centerdot 4}=5^{4\centerdot \frac{1}{3}}=\left( 5^{4} \right)^{\frac{1}{3}}=\left( 5\centerdot 5\centerdot 5\centerdot 5 \right)^{\frac{1}{3}}=625^{\frac{1}{3}}.

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