<s>[INST] How do I express 108 as the product of powers of its prime factors? How do I express $108$ as the product of powers of its prime factors?
I've got $2^2 \times 3^3$ is this correct?
 [/INST] First notice that $108$ is even so $2$ is a factor:
$$\begin{matrix} & & 108 \\ & \large\swarrow & & \large\searrow \\ 2 & &  & &54\end{matrix}$$
$54$ is also even:
$$\begin{matrix} & & 54 \\ & \large\swarrow & & \large\searrow \\ 2 & &  & &27\end{matrix}$$
$27$ is not even so let's check the next prime: $3$.  We can see that $3$ divides $27$ because $2+7=9$ is a multiple of $3$.  So:
$$\begin{matrix} & & 27 \\ & \large\swarrow & & \large\searrow \\ 3 & &  & &9\end{matrix}$$
And then hopefully you can see that $9$'s prime factorization is just $3\times 3$.  So putting all of these steps together we see that
$$\begin{matrix} & & 108 \\ & \large\swarrow & & \large\searrow  \\ \color{red}2 & & & & 54 \\ & & & \large\swarrow & & \large\searrow \\ & & \color{red}2 & & & & 27 \\ & & & & & \large\swarrow & & \large\searrow \\ & & & & \color{red}3 & & & & 9 \\ & & & & & & & \large\swarrow & & \large\searrow \\ & & & & & & \color{red}3 & & & & \color{red}3\end{matrix}$$
Thus 
$$108 = 2\times 2 \times 3 \times 3\times 3 = 2^2\times 3^3$$
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