Aus Online Mathematik Brückenkurs 1

We can factorize the exponents \displaystyle 40 and \displaystyle 56 as

\displaystyle \begin{align} & 40=4\centerdot 10=2\centerdot 2\centerdot 2\centerdot 5=2^{3}\centerdot 5 \\ & \\ & 56=7\centerdot 8=7\centerdot 2\centerdot 4=7\centerdot 2\centerdot 2\centerdot 2=2^{3}\centerdot 7 \\ \end{align}

and we then see that they have \displaystyle 2^{3}=8 as a common factor. We can take this factor out as an "outer" exponent:

\displaystyle \begin{align} & 3^{40}=3^{5\centerdot 8}=\left( 3^{5} \right)^{8}=\left( 3\centerdot 3\centerdot 3\centerdot 3\centerdot 3 \right)^{8}=243^{8} \\ & \\ & 2^{56}=2^{7\centerdot 8}=\left( 2^{7} \right)^{8}=\left( 2\centerdot 2\centerdot 2\centerdot 2\centerdot 2\centerdot 2\centerdot 2 \right)^{8}=128^{8} \\ \end{align}

This shows that \displaystyle 3^{40}=243^{8} is bigger than \displaystyle 2^{56}=128^{8}