Why does any non-zero number to the zero power equal one?

The zero exponent rule follows from the Quotient Rule for Exponents: \frac{a^n}{a^n} = a^{n-n} = a^0. Since any number divided by itself equals one, we get a^0 = 1. This definition keeps exponent rules consistent.

What happens if the base is zero?

0^0 is undefined in most algebra courses. The zero exponent rule only applies to non-zero bases. If you encounter 0^0 in a problem, you cannot simplify it using this rule.

Does the zero exponent rule work for negative bases?

Yes, negative numbers follow the same rule: (-2)^0 = 1, (-100)^0 = 1, and so on. The zero exponent 'turns off' the negative sign and gives a result of one.

How is this different from multiplying by zero?

Multiplying by zero gives zero (5 \cdot 0 = 0), but raising to the zero power gives one (5^0 = 1). These are completely different operations: zero as an exponent is not the same as zero as a multiplier.

Zero Exponent

Any non-zero number raised to the power of zero always equals one, providing a key rule for simplifying exponential expressions and solving equations.

a^0 = 1, \quad a \neq 0

Solve a problem with Zero Exponent

Type the problem. The solver will use Zero Exponent where Zero Exponent is the right tool, and tell you when it is not.

What each symbol means

What Zero Exponent takes

Symbol	Meaning
$a$	Any non-zero real number (the base); zero to the zero power is undefined in most algebra contexts, and $a$ must be non-zero for this rule to apply.

When to use it

Use this rule whenever you need to evaluate or simplify an expression where a non-zero base is raised to the zero power.

Level

Usually taught in: Algebra I

Worked examples

1. Simplify an arithmetic expression with a zero exponent

Problem

Simplify

9 + 10^0 - 1

$10^0 = 1$
Evaluate the term with the zero exponent.
$9 + 1 = 10$
Add $9$ and the result.
$10 - 1 = 9$
Subtract $1$ to get the final answer.

Answer: $9$

We apply the zero exponent rule to simplify $10^0 = 1$ . Then we perform the arithmetic operations in order: add $9 + 1 = 10$ , then subtract $1$ to get $9$ .

2. Simplify with a negative base and zero exponent

Problem

Simplify

2 \cdot (-4)^0 + 5

$(-4)^0 = 1$
Apply the zero exponent rule; even though the base is negative, the result is still one.
$2 \cdot 1 = 2$
Multiply $2$ by the result from step 1.
$2 + 5 = 7$
Add the constant term to get the final answer.

Answer: $7$

Negative bases still follow the zero exponent rule. We evaluate $(-4)^0 = 1$ , then multiply and add to get $7$ .

3. Calculate distance on a road trip with two driving legs

Problem

Alex is taking a road trip. On the first leg, she drives at 60 mph for

2^0

hours. On the second leg, she drives at 50 mph for

3^0

hours. How far did she drive in total? (Use distance = rate × time.)

$2^0 = 1$
Evaluate the zero exponent for the first leg's time.
$60 \cdot 1 = 60$
Calculate distance for the first leg using distance = rate × time.
$3^0 = 1$
Evaluate the zero exponent for the second leg's time.
$50 \cdot 1 = 50$
Calculate distance for the second leg.
$60 + 50 = 110$
Add the distances from both legs to find the total.

Answer: $110 \text{ miles}$

We use the zero exponent rule to simplify both times to 1 hour. Then we apply the distance formula to each leg and add them together to get 110 miles total.

Common mistakes

Where Zero Exponent usually goes wrong

The mistake	Why it is wrong	The fix
$5^0 = 0$	Zero as an exponent doesn't make the base disappear; the rule states that any non-zero base to the zero power equals one.	$5^0 = 1$ , not $0$ .
$0^0 = 1$	Zero to the zero power is undefined in most algebra courses because there is no universally agreed value.	The zero exponent rule only applies when the base is non-zero; $0^0$ is undefined.
$(-5)^0 = -1$	Negative bases still follow the zero exponent rule; the exponent zero applies to the entire base, not just to its absolute value.	$(-5)^0 = 1$ , not $-1$ .

Tips and when to use something else

Whenever you see any non-zero number (except zero itself) raised to the power of zero, replace it with one.
This rule works for any non-zero base: positive integers, negative integers, fractions, decimals, and variables.
If you're solving an equation and encounter a term like $x^0$ , remember it simplifies to $1$ (assuming $x \neq 0$ ), which often helps isolate the variable.
The zero exponent rule is a direct consequence of the Quotient Rule for Exponents: when you divide $a^n$ by $a^n$ , you get $a^{n-n} = a^0$ , and since $\frac{a^n}{a^n} = 1$ , this is why $a^0 = 1$ .

Related formulas

Quotient Rule for ExponentsQuotient Rule for Exponents: when dividing powers with the same base, subtract the exponents to simplify the expression.Product Rule for ExponentsThe Product Rule for Exponents simplifies multiplication of powers with the same base by adding the exponents into a single power.Negative ExponentA negative exponent tells you to take the reciprocal: a^{-n} equals 1/a^n. Use this to rewrite expressions with negative powers.Power Rule for ExponentsSimplify nested exponents by multiplying the exponents together: this is the fundamental tool you need whenever one exponent is itself raised to a power.

Frequently asked questions

Why does any non-zero number to the zero power equal one?: The zero exponent rule follows from the Quotient Rule for Exponents: $\frac{a^n}{a^n} = a^{n-n} = a^0$ . Since any number divided by itself equals one, we get $a^0 = 1$ . This definition keeps exponent rules consistent.
What happens if the base is zero?: $0^0$ is undefined in most algebra courses. The zero exponent rule only applies to non-zero bases. If you encounter $0^0$ in a problem, you cannot simplify it using this rule.
Does the zero exponent rule work for negative bases?: Yes, negative numbers follow the same rule: $(-2)^0 = 1$ , $(-100)^0 = 1$ , and so on. The zero exponent 'turns off' the negative sign and gives a result of one.
How is this different from multiplying by zero?: Multiplying by zero gives zero ( $5 \cdot 0 = 0$ ), but raising to the zero power gives one ( $5^0 = 1$ ). These are completely different operations: zero as an exponent is not the same as zero as a multiplier.

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18