Problem:
Compute the geometric mean of ${125, 1, 27, 16, 1}$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the geometric mean of the list}: \\
 (125,1,27,16,1)\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{= }\text{125 }\text{$\times $ }1 \text{$\times $ }\text{27 }\text{$\times $ }\text{16 }\text{$\times $ }1: \\
 \sqrt[(\text{number of elements})]{(\text{125 }\text{$\times $ }1 \text{$\times $ }\text{27 }\text{$\times $ }\text{16 }\text{$\times $ }1)} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Counting, }\text{we }\text{see }\text{that }\text{the }\text{list }\text{has }5 \text{elements}: \\
 (\text{125 }\text{$\times $ }1 \text{$\times $ }\text{27 }\text{$\times $ }\text{16 }\text{$\times $ }1)^{\frac{1}{5}} \text{= }\sqrt[5]{\text{125 }\text{$\times $ }1 \text{$\times $ }\text{27 }\text{$\times $ }\text{16 }\text{$\times $ }1} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{125 }\text{$\times $ }1 \text{$\times $ }\text{27 }\text{$\times $ }\text{16 }\text{$\times $ }1=54000: \\
 \sqrt[5]{54000} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt[5]{54000} \text{= }\sqrt[5]{2^4\times 15^3} \text{= }2^{4/5}\times 15^{3/5}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 2^{4/5}\times 15^{3/5} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/counting_and_statistics/geometric_mean_w_steps/4033.txt'}