Zero and Negative Exponents: Why a⁰ = 1 and a⁻ⁿ = 1/aⁿ

Q: Does a negative exponent make the number negative?

No. It makes a reciprocal. 5^-2 = 1/25, which is positive.

Two definitions that look like magic until you follow the pattern of powers downwards. This article shows why $a^0 = 1$ , what a negative exponent really means, and the common slip of reading $3^{-2}$ as $-9$ .

3 min read Supports 3 lessons

$2^3$ means three $2$ s multiplied together. But what could $2^0$ , zero $2$ s multiplied together, possibly mean? Or $2^{-1}$ , negative one $2$ s? The definitions come from asking what makes the pattern of powers carry on smoothly.

Follow the pattern down

Each step down in the exponent divides by the base. Keep going past $2^1$ and the pattern tells you what comes next.

$2^3 = 8$
Start high.

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The graph of $2^x$ passes smoothly through $(0, 1)$ and stays above zero for negative $x$ .

Zero and negative exponents

a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad (a \neq 0)

The same answer from the exponent laws

The quotient law agrees. Any number divided by itself is $1$ , and the law says it is also $a^0$ :

\frac{a^3}{a^3} = 1 \qquad\text{and}\qquad \frac{a^3}{a^3} = a^{3 - 3} = a^0 \qquad\Rightarrow\qquad a^0 = 1

\frac{a^2}{a^5} = \frac{1}{a^3} \qquad\text{and}\qquad \frac{a^2}{a^5} = a^{2 - 5} = a^{-3}

Key idea

These are not new rules to memorise. They are the only definitions that keep every exponent law working.

Common mistakes

Watch out

A negative exponent makes a reciprocal, not a negative number. $3^{-2} = \frac{1}{9}$ , not $-9$ . The answer is still positive.

Watch out

In $2x^{-1}$ , only the $x$ has the negative power. It means $\frac{2}{x}$ , not $\frac{1}{2x}$ .

Negative powers of fractions

Since a negative exponent flips the base over, a fraction to a negative power flips and then takes the positive power.

\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}

Worked example

Simplify $\dfrac{4x^{-2}}{x^{3}}$ and write it with positive exponents.

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Quotient law.
$4x^{-2 - 3} = 4x^{-5}$

Where you will see them

Negative powers of ten write small numbers in scientific notation: $10^{-3} = 0.001$ . In calculus, writing $\frac{1}{x^2}$ as $x^{-2}$ lets you use the same rules for every power.

Watch out

$0^0$ and $0^{-1}$ are left undefined here: you cannot divide by zero.

Common questions

Why is any number to the power of zero equal to 1?: Because each step down in the exponent divides by the base. From $a^1 = a$ , one more step gives $a \div a = 1$ . The quotient law $\frac{a^n}{a^n} = a^0$ gives the same answer.
Does a negative exponent make the number negative?: No. It makes a reciprocal. $5^{-2} = \frac{1}{25}$ , which is positive.

Practise zero and negative exponents

The Zero & Negative Exponents lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Why is any number to the power of zero equal to 1?

Because each step down in the exponent divides by the base. From $a^1 = a$ , one more step gives $a \div a = 1$ . The quotient law $\frac{a^n}{a^n} = a^0$ gives the same answer.

Does a negative exponent make the number negative?

No. It makes a reciprocal. $5^{-2} = \frac{1}{25}$ , which is positive.