Problem:
Compute the geometric mean of ${4, 125, 81, 512}$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the geometric mean of the list}: \\
 (4,125,81,512)\text{} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{The }\text{geometric }\text{mean }\text{of }\text{a }\text{list }\text{of }\text{numbers }\text{is }\text{given }\text{by}: \\
 (\text{product of elements})^{\frac{1}{(\text{number of elements})}} \\
\end{array}
 \\
 
\begin{array}{l}
 (\text{product of elements}) \text{= }4 \text{$\times $ }\text{125 }\text{$\times $ }\text{81 }\text{$\times $ }512: \\
 \sqrt[(\text{number of elements})]{(4 \text{$\times $ }\text{125 }\text{$\times $ }\text{81 }\text{$\times $ }512)} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Counting, }\text{we }\text{see }\text{that }\text{the }\text{list }\text{has }4 \text{elements}: \\
 (4 \text{$\times $ }\text{125 }\text{$\times $ }\text{81 }\text{$\times $ }512)^{\frac{1}{4}} \text{= }\sqrt[4]{4 \text{$\times $ }\text{125 }\text{$\times $ }\text{81 }\text{$\times $ }512} \\
\end{array}
 \\
 
\begin{array}{l}
 4 \text{$\times $ }\text{125 }\text{$\times $ }\text{81 }\text{$\times $ }512=20736000: \\
 \sqrt[4]{20736000} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt[4]{20736000} \text{= }\sqrt[4]{2^{11}\times 3^4\times 5^3} \text{= }2^2\times 3\times 2^{3/4}\times 5^{3/4}: \\
 2^2\times 3\times 2^{3/4}\times 5^{3/4} \\
\end{array}
 \\
 
\begin{array}{l}
 2^2=4: \\
 \fbox{$4$}\times 3\times 2^{3/4}\times 5^{3/4} \\
\end{array}
 \\
 
\begin{array}{l}
 4\times 3 \text{= }12: \\
 \fbox{$12$}\times 2^{3/4}\times 5^{3/4} \\
\end{array}
 \\
 
\begin{array}{l}
 2^{3/4}\times 5^{3/4}=(2\times 5)^{3/4}: \\
 12 \fbox{$(2\times 5)^{3/4}$} \\
\end{array}
 \\
 
\begin{array}{l}
 2\times 5 \text{= }10: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 12\times \fbox{$10$}^{3/4} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/counting_and_statistics/geometric_mean_w_steps/4514.txt'}