Negative law of exponents

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The negative law of exponents says x−a=1xax^{-a} = \frac{1}{x^a} for any nonzero xx.
A negative exponent is an instruction to take the reciprocal, not to make the number negative.
negative_exponents.png
So a negative power moves the base to the denominator.
4−2=142=1164^{-2} = \frac{1}{4^2} = \frac{1}{16}, and 3−1=133^{-1} = \frac{1}{3}.
The result is a positive fraction, since the sign of the base is untouched.
The rule also runs in reverse: 1x−a=xa\frac{1}{x^{-a}} = x^a.
A base with a negative exponent sitting in a denominator moves up to the top with a positive exponent.
So 13−2=32=9\frac{1}{3^{-2}} = 3^2 = 9, and 1(2x)−2=(2x)2\frac{1}{(2x)^{-2}} = (2x)^2.
For a fraction raised to a negative power, flip the fraction.
Since (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n, we get (53)−2=(35)2\left(\frac{5}{3}\right)^{-2} = \left(\frac{3}{5}\right)^2.
This is handy for exponential expressions: (23)−x=(32)x\left(\frac{2}{3}\right)^{-x} = \left(\frac{3}{2}\right)^x.

Worked examples

Evaluate 2−32^{-3}.
A negative exponent takes the reciprocal: 2−3=1232^{-3} = \frac{1}{2^3}.
So it is 18\frac{1}{8}.
Simplify 15−2\frac{1}{5^{-2}}.
A negative exponent in the denominator moves up top: 15−2=52\frac{1}{5^{-2}} = 5^2.
So it is 2525.
Rewrite (34)−2\left(\frac{3}{4}\right)^{-2} without a negative exponent.
Flip the fraction and drop the negative: (34)−2=(43)2\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2.
So it is 169\frac{16}{9}.

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