The Negative Exponent Rule states that for any non-zero real number $a$ and any integer $n$:
A negative exponent tells you to take the reciprocal of the base. It’s like a ticket that moves the number from the “upstairs” (numerator) to the “downstairs” (denominator), or vice versa. Once you move it, the exponent becomes positive.
Math is all about patterns. To understand why negative exponents result in fractions, let’s look at powers of 3:
$3^3 = 27$
$3^2 = 9$ (We divided by 3)
$3^1 = 3$ (We divided by 3 again)
$3^0 = 1$ (We divided by 3 again—remember, anything to the power of 0 is 1!)
Now, what happens if we keep dividing by 3?
$3^{-1} = \frac{1}{3}$
$3^{-2} = \frac{1}{9}$ (which is $\frac{1}{3^2}$)
As you can see, as the exponent goes down by 1, we divide the value by the base. Crossing into negative territory just means we start creating fractions.
When you see a negative exponent, follow these two steps:
Flip the Base: Move the base to the other side of the fraction line.
If it’s on top (numerator), move it to the bottom.
If it’s on the bottom (denominator), move it to the top.
Change the Sign: Make the exponent positive.
Problem: Simplify $4^{-3}$
Flip it: Move $4$ to the denominator: $\frac{1}{4…}$
Change sign: Turn $-3$ into $+3$: $\frac{1}{4^3}$
Solve: $\frac{1}{64}$
Problem: Simplify $x^{-5}$
What if the negative exponent is already in the denominator?
Problem: $\frac{1}{y^{-4}}$
Think of the negative sign as a request to “move.” Since $y$ is already on the bottom, it wants to move to the top!
Mistake #1: Making the result negative.
Wrong: $5^{-2} = -25$
Right: $5^{-2} = \frac{1}{25}$
Remember: A negative exponent does not affect the sign of the base number. It only affects the location (numerator vs. denominator).
Mistake #2: Flipping the coefficient.
Problem: $2x^{-3}$
The exponent $-3$ belongs only to the $x$, not the $2$.
Right: $\frac{2}{x^3}$ (The 2 stays on top!)
| Expression | Action | Result |
| $x^{-n}$ | Move to denominator | $\frac{1}{x^n}$ |
| $\frac{1}{x^{-n}}$ | Move to numerator | $x^n$ |
| $(\frac{a}{b})^{-n}$ | Flip the whole fraction | $(\frac{b}{a})^n$ |
Ready to test your skills? Try simplifying this expression:
(Scroll down for the answer!)
Answer: $\frac{3b^4}{a^2}$
(The $a$ moves down, the $b$ moves up, and the $3$ stays put!)