<s>[INST] Why does $2^{-n} = 5^n \times 10^{-n}$? If we look at the decimal equivilents of $2^{-n}$, we see they resemble $5^n$ with a decimal point in front of them:
$\begin{align}
2^{-1} &= 0.5 \\
2^{-2} &= 0.25 \\
2^{-3} &= 0.125 \\
2^{-4} &= 0.0625 \\
2^{-5} &= 0.03125 \\
...
\end{align}$
It looks like it's as simple as saying $2^{-n} = 5^n \times 10^{-n}$, and when we calculate that out, it's correct:
$\begin{align}
5^1 \times 10^{-1} &= 5 \times 0.1 = 0.5 \\
5^2 \times 10^{-2} &= 25 \times 0.01 = 0.25 \\
5^3 \times 10^{-3} &= 125 \times 0.001 = 0.125 \\
5^4 \times 10^{-4} &= 625 \times 0.0001 = 0.0625 \\
5^5 \times 10^{-5} &= 3125 \times 0.00001 = 0.03125 \\
...
\end{align}$
I calculated this out for $n = [0, 10]$ and it works out, but I have no idea how to prove it fully.
 [/INST] $\dfrac{1}{2^x}= \dfrac{5^x\cdot2^x}{2^x\cdot10^x}=\dfrac{5^x}{10^x}$
</s>