Aus Online Mathematik Brückenkurs 1

The numbers \displaystyle 9 and \displaystyle 27 can both be written as powers of \displaystyle 3,

\displaystyle \begin{align} & 9=3\centerdot 3=3^{2} \\ & \\ & 27=3\centerdot 9=3\centerdot 3\centerdot 3=3^{3} \\ \end{align}

Thus, all factors in the expression can be written using a common base

and the whole product can be simplified using the power rules

\displaystyle \begin{align} & 3^{13}\centerdot 9^{-3}27^{-2}=3^{13}\centerdot \left( 3^{2} \right)^{-3}\centerdot \left( 3^{3} \right)^{-2} \\ & \\ & =3^{13}\centerdot 3^{2\centerdot \left( -3 \right)}\centerdot 3^{3\centerdot \left( -2 \right)}=3^{13}\centerdot 3^{-6}\centerdot 3^{-6} \\ & \\ & =3^{13-6-6}=3^{1}=3 \\ \end{align}