Zero & Negative Exponents

Q: Why does anything to the zero power equal 1?

Follow the pattern: each time the exponent drops by $1$, you divide by the base. $2^{2} = 4$, $2^{1} = 2$, so $2^{0}$ must be $2 \div 2 = 1$. It also keeps the quotient rule consistent: $\dfrac{2^{3}}{2^{3}} = 2^{0}$, and a number divided by itself is $1$.

Exponents of $0$ and negative exponents look strange at first — how do you multiply a number by itself zero times, or negative-two times? The rules turn out to be simple. Any nonzero number to the $0$ power equals $1$ : $2^{0} = 1$ and $517^{0} = 1$ . And a negative exponent means take the reciprocal, then make the exponent positive: $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$ .

One warning before anything else: a negative exponent never makes the answer negative. $3^{-2}$ is $\dfrac{1}{9}$ , a small positive fraction — not $-9$ and not $-\dfrac{1}{9}$ . The negative sign in the exponent is an instruction to flip, nothing more.

The zero power

Any nonzero number raised to the $0$ power equals $1$ . You can see why by walking a pattern downward: $2^{3} = 8$ , $2^{2} = 4$ , $2^{1} = 2$ . Each time the exponent drops by $1$ , the value gets divided by the base. One more step down from $2^{1} = 2$ gives $2^{0} = 2 \div 2 = 1$ .

The base can be anything except $0$ : $9^{0} = 1$ , $517^{0} = 1$ , and $\left(\dfrac{3}{4}\right)^{0} = 1$ . The size of the base is irrelevant — the zero power flattens everything to $1$ .

Negative exponents: flip it

Keep the same pattern going below zero. After $2^{0} = 1$ , dividing by $2$ again gives $2^{-1} = \dfrac{1}{2}$ , then $2^{-2} = \dfrac{1}{4}$ , then $2^{-3} = \dfrac{1}{8}$ . A negative exponent lands you on a fraction: the reciprocal of the matching positive power.

As a rule: $2^{-a} = \dfrac{1}{2^{a}}$ . To evaluate, flip the base into the bottom of a fraction and make the exponent positive. So $3^{-2} = \dfrac{1}{3^{2}} = \dfrac{1}{9}$ — positive, because you are dividing by a positive number, not subtracting anything.

A negative exponent in the denominator

Flipping works in both directions across the fraction bar. If the negative exponent is already in the bottom, flipping sends the power to the top: $\dfrac{1}{2^{-3}} = 2^{3} = 8$ .

A quick way to think about it: a negative exponent means the power is on the wrong side of the fraction bar. Move it to the other side and the exponent turns positive.

Worked examples

Example 1: the zero power

Evaluate $9^{0}$ .

The base is nonzero, so the zero-power rule applies

9^{0} = 1

Answer: $1$

Example 2: a negative exponent

Write $2^{-3}$ as a fraction.

Flip the base and make the exponent positive

2^{-3} = \dfrac{1}{2^{3}}

Evaluate the power

2^{3} = 8

Write the result

2^{-3} = \dfrac{1}{8}

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

Example 3: negative exponent in the bottom

Evaluate $\dfrac{1}{5^{-2}}$ .

The power is on the wrong side — move it up

\dfrac{1}{5^{-2}} = 5^{2}

Evaluate

5^{2} = 25

Answer: $25$

Try one yourself

Common questions

Does a negative exponent make the answer negative?

No. A negative exponent means reciprocal, not negative. $3^{-2} = \dfrac{1}{9}$ , which is positive. If your answer to a negative-exponent problem has a negative sign, go back and find where it crept in.

Why does anything to the zero power equal 1?

Follow the pattern: each time the exponent drops by $1$ , you divide by the base. $2^{2} = 4$ , $2^{1} = 2$ , so $2^{0}$ must be $2 \div 2 = 1$ . It also keeps the quotient rule consistent: $\dfrac{2^{3}}{2^{3}} = 2^{0}$ , and a number divided by itself is $1$ .

What about

0^{0}

The zero-power rule only covers nonzero bases. $0^{0}$ is left undefined in this course — you will not be asked to evaluate it, and no rule in this unit applies to it.

Where do negative exponents actually get used?

Small numbers in scientific notation. A number like $0.001$ is written $10^{-3}$ , and quantities like the width of a cell or a grain of pollen use negative powers of $10$ . The flip rule is exactly what makes those conversions work.

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Zero & Negative Exponents

The zero power

Negative exponents: flip it

A negative exponent in the denominator

Worked examples

Example 1: the zero power

Example 2: a negative exponent

Example 3: negative exponent in the bottom

Try one yourself

Common questions

Keep learning

Exponent Notation & Meaning

Power of a Power

Writing Numbers in Scientific Notation

Operations in Scientific Notation