Earlier we focused on powers of 10 because 10 plays a special role in the decimal number system. But the exponent rules that we developed for 10 also work for other bases. For example, if $2^0=1$ and $2^{\text -n} = \frac{1}{2^n}$, then

\(\begin{align}2^m \boldcdot 2^n &= 2^{m+n} \\ \left(2^m\right)^n &= 2^{m \boldcdot n} \\ \frac{2^m}{2^n} &= 2^{m\text -n}. \end{align}\)

These rules also work for powers of numbers less than 1. For example, $\left(\frac{1}{3}\right)^2 = \frac{1}{3} \boldcdot \frac{1}{3}$ and $\left(\frac{1}{3}\right)^4 = \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3}$. We can also check that $\left(\frac{1}{3}\right)^2 \boldcdot \left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^{2+4}$.

Using a variable $x$ helps to see this structure. Since $x^2 \boldcdot {x^5} = x^7$ (both sides have 7 factors that are $x$), if we let $x = 4$, we can see that $4^2 \boldcdot 4^5 = 4^7$. Similarly, we could let $x = \frac{2}{3}$ or $x = 11$ or any other positive value and show that these relationships still hold.