AI Tutor
If zooming an image by a factor of 232^3 doubles its width three times to make it 8 times larger, what does zooming by 2−32^{-3} do?
It shrinks the image by cutting it in half three times, producing 18\frac{1}{8} of the original size.
In 1655, English mathematician John Wallis systematically used negative powers to maintain this exact pattern: multiplying adds exponents, so moving backward must subtract them.
A negative exponent instructs you to take the reciprocal (the inverted fraction 1a\frac{1}{a}) of the base: for any non-zero base aa, a−n=1ana^{-n} = \frac{1}{a^n}.
Why does this hold algebraically? Dividing powers with the same base gives a0an=a0−n=a−n\frac{a^0}{a^n} = a^{0-n} = a^{-n}, and since a0=1a^0 = 1, a0an\frac{a^0}{a^n} is identically 1an\frac{1}{a^n}.
What happens when a negative exponent sits next to an ordinary number?

The Base Trap

A coefficient is a numerical factor placed directly before a variable, like the 55 in 5x−35x^{-3}.
Unless parentheses group them together, an exponent applies only to the base immediately to its left.
Consider the expression −4a−2b3-4a^{-2}b^3. The coefficient −4-4 and the variable b3b^3 have positive exponents ((−4)1(-4)^1 and b3b^3), so only a−2a^{-2} moves downstairs: −4a−2b3=−4b3a2-4a^{-2}b^3 = \frac{-4b^3}{a^2}.
Notice that the negative sign on −4-4 stays negative; negative exponents change position across a fraction bar, not the numerical sign of a coefficient.