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Howdy! Here are a few very handy rules of algebra. These basic rules are useful for everything from figuring out your gas mileage to acing your next math test — or even solving equations from the far reaches of theoretical physics. Happy calculating!

Algebra Rule 14

The result of a negative exponent is the inverse of the same positive exponent

$$a^{-n} = {1\over a^n}$$
Description:

It might seem odd to have a negative exponent (since you can't multiply something by itself a negative number of times). However, if we take a closer look at the rule $a^na^m = a^{n+m}$ we can see that it implies that $a^{-n}$ must equal ${1 \over a^n}$, the multiplicative inverse or reciprocal of $a^n$.

This becomes clear looking at the $a^{n+m}$ side of the equation from rule 11. What happens if $m$ is negative? Obviously, this will reduce the combined value of the exponent (for example, $2^{4-2} = 2^2$). What does this mean for the left hand side of the $a^na^m = a^{n+m}$ equation? It means that the value of, for example, $2^4$ must be reduced to $2^2$ when it is multiplied by $2^{-2}$. If, as this rule states, $a^{-n} = {1 \over a^n}$, this works out perfectly: $2^4 * 2^{-2} = 2^4 * {1 \over 2^2} = 16 * {1 \over 4} = 4 = 2^2 = 2^{4-2}$

$$2^{-2} = {1\over2^2} = {1\over 4}$$
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