Negative Exponent Rule

Q: Does a negative exponent make the answer negative?

No. $2^{-2} = \dfrac{1}{4}$, which is positive. The negative sign lives in the exponent and means reciprocal. The sign of the answer comes from the base, not the exponent.

Q: What happens to a negative exponent in the denominator?

It moves up. $\dfrac{1}{x^{-2}} = x^{2}$. Crossing the fraction bar in either direction flips the exponent's sign.

Q: In $3x^{-2}$, does the 3 move down too?

No. The exponent $-2$ belongs only to $x$, so only $x^{2}$ moves: $3x^{-2} = \dfrac{3}{x^{2}}$. The $3$ has no negative exponent, so it stays where it is.

A negative exponent does not make anything negative. It tells you to take the reciprocal — move the factor across the fraction bar and make the exponent positive: $x^{-4} = \dfrac{1}{x^{4}}$ .

That surprises a lot of students, so check it with numbers: $2^{-2}$ is $\dfrac{1}{2^{2}} = \dfrac{1}{4}$ , a small positive number — not $-4$ . Negative exponents come from the division rule: keep subtracting exponents past zero and you land below it, which just means the leftover factors live on the bottom of the fraction.

The rule

In symbols, $x^{-a} = \dfrac{1}{x^{a}}$ . The negative sign in the exponent means flip: move the power to the other side of the fraction bar and drop the negative.

It works in both directions. A negative exponent in a denominator moves the power up: $\dfrac{1}{x^{-2}} = x^{2}$ . Either way, crossing the fraction bar flips the sign of the exponent.

Where negative exponents come from

Look at the pattern of powers of $2$ : $2^{3} = 8$ , $2^{2} = 4$ , $2^{1} = 2$ , $2^{0} = 1$ . Each step down divides by $2$ . Keep going: $2^{-1} = \dfrac{1}{2}$ , $2^{-2} = \dfrac{1}{4}$ . Negative exponents continue the pattern below zero — they're fractions, not negatives.

You'll also meet them through the division rule: $\dfrac{x^{2}}{x^{5}} = x^{2-5} = x^{-3}$ . Three $x$ 's survive the cancelling on the bottom, which is exactly $\dfrac{1}{x^{3}}$ .

Only the power moves

A coefficient without a negative exponent stays put. In $3x^{-2}$ , only the $x^{2}$ crosses the bar: $\dfrac{3}{x^{2}}$ , not $\dfrac{1}{3x^{2}}$ . Move exactly the factors whose exponents are negative and nothing else.

Worked examples

Example 1: rewrite with a positive exponent

Rewrite $x^{-3}$ with a positive exponent.

Negative exponent means take the reciprocal

x^{-3} = \dfrac{1}{x^{3}}

Answer: $\dfrac{1}{x^{3}}$

Example 2: evaluate with numbers

Evaluate $2^{-2}$ .

Flip and make the exponent positive

2^{-2} = \dfrac{1}{2^{2}}

Evaluate the power

\dfrac{1}{4}

Answer: $\dfrac{1}{4}$

Example 3: combine with the division rule

Simplify $\dfrac{x^{-5}}{x^{3}}$ .

Division rule: subtract the exponents

x^{-5-3} = x^{-8}

Rewrite with a positive exponent

\dfrac{1}{x^{8}}

Answer: $\dfrac{1}{x^{8}}$

Try one yourself

Common questions

Does a negative exponent make the answer negative?

No. $2^{-2} = \dfrac{1}{4}$ , which is positive. The negative sign lives in the exponent and means reciprocal. The sign of the answer comes from the base, not the exponent.

What happens to a negative exponent in the denominator?

It moves up. $\dfrac{1}{x^{-2}} = x^{2}$ . Crossing the fraction bar in either direction flips the exponent's sign.

3x^{-2}

, does the 3 move down too?

No. The exponent $-2$ belongs only to $x$ , so only $x^{2}$ moves: $3x^{-2} = \dfrac{3}{x^{2}}$ . The $3$ has no negative exponent, so it stays where it is.

How is this different from the zero power?

$x^{0} = 1$ for any nonzero base — that's where the pattern crosses from positive powers to negative ones. Negative exponents pick up right after: $x^{-1} = \dfrac{1}{x}$ , $x^{-2} = \dfrac{1}{x^{2}}$ , and so on.

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Negative Exponent Rule

The rule

Where negative exponents come from

Only the power moves

Worked examples

Example 1: rewrite with a positive exponent

Example 2: evaluate with numbers

Example 3: combine with the division rule

Try one yourself

Common questions

Keep learning

Multiplication Rule

Division Rule

Power Rule

Power of a Fraction and Rational Exponents