At this level $0^0$ is left undefined. The pattern argument needs us to divide by the base, and you cannot divide by $0$. That is why the rule is written $a^0 = 1$ for $a \neq 0$.

Does the rule work for negative numbers and fractions?

Yes, as long as the base is not zero and the whole base is in brackets: $(-8)^0 = 1$ and $\left(\frac{3}{4}\right)^0 = 1$.

Why does my calculator give $-1$ for $-3^0$?

The calculator follows the order of operations: the power comes before the minus sign, so it computes $-(3^0) = -1$. Type $(-3)^0$ with brackets to get $1$.

Teacher Resources

Zero Exponent

Learn why any nonzero number to the power of zero equals 1, and how to handle expressions like $5x^0$ and $(-3)^0$.

Intermediate20 minLesson

Printable worksheet

Definition

The zero exponent rule says that any nonzero number raised to the power of

0

equals

1

a^0 = 1 \quad (a \neq 0)

7^0 = 1

100^0 = 1

(-5)^0 = 1

and

\left(\frac{2}{3}\right)^0 = 1

Why? Look at the pattern. Every time the exponent goes down by

1

, you divide by the base:

Power	Value
$2^4$	$16$
$2^3$	$8$
$2^2$	$4$
$2^1$	$2$
$2^0$	$2 \div 2 = 1$

The quotient rule agrees. Dividing a power by itself gives

1

, and the quotient rule gives exponent

0

\frac{a^n}{a^n} = a^{n-n} = a^0 \quad \text{and} \quad \frac{a^n}{a^n} = 1

Watch what the exponent belongs to. An exponent only applies to the base directly in front of it:

$5x^0 = 5 \times x^0 = 5 \times 1 = 5$ , but $(5x)^0 = 1$
$-3^0 = -(3^0) = -1$ , but $(-3)^0 = 1$

Special case:

0^0

is left undefined at this level. The rule

a^0 = 1

is only for

a \neq 0

Try it now

What is

9^0

Worked Examples

Use the pattern of powers of $3$ to find $3^0$ .

Start with a power you know

$3^3 = 3 \times 3 \times 3$ → $3^3 = 27$

Lower the exponent by 1: divide by 3

$27 \div 3 = 9$ → $3^2 = 9$

Lower it again: divide by 3

$9 \div 3 = 3$ → $3^1 = 3$

One more step down to exponent 0

$3 \div 3 = 1$ → $3^0 = 1$

Answer: $3^0 = 1$

Common Mistakes

Writing $a^0 = 0$ , for example $9^0 = 0$

Why it's wrong: The exponent $0$ does not mean multiplying by zero. Going down the pattern means dividing by the base, and $9 \div 9 = 1$ .

Correct: $9^0 = 1$ , because $\frac{9^1}{9^1} = 9^0$ and $\frac{9}{9} = 1$

Writing $a^0 = a$ , for example $9^0 = 9$

Why it's wrong: That is the rule for exponent $1$ : $9^1 = 9$ . Exponent $0$ is one more step down, so you divide by $9$ once more.

Correct: $9^1 = 9$ but $9^0 = 1$

Writing $5x^0 = 1$

Why it's wrong: The exponent belongs only to $x$ , not to the coefficient $5$ . Only $x^0$ becomes $1$ .

Correct: $5x^0 = 5 \times 1 = 5$ , while $(5x)^0 = 1$

Writing $-3^0 = 1$

Why it's wrong: Without brackets, the base is $3$ , not $-3$ . The minus sign stays in front of the result.

Correct: $-3^0 = -(3^0) = -1$ , while $(-3)^0 = 1$

Interactive Visual

Exponents as repeated factors

Quotient rule

Same base, divide: subtract the exponents.

\frac{5^{3}}{5^{3}}

Write each power as repeated factors

5

5

5

5

5

5

3 common factors cancel, top and bottom

All factors cancel: the result is 1, so the exponent is 0

\displaystyle 1 = 5^{0}

Result

\displaystyle \frac{5^{3}}{5^{3}}

=

\displaystyle 5^{3 - 3}

=

\displaystyle 5^{0}

=

\displaystyle 1

Value: $5^{0} = 1$

This works for every base except 0

Base:

Calculate:2³ = 8

See how powers of a number grow on the number line. Change the base and exponent.

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What is $9^0$ ?

Why It Matters

The zero exponent turns up wherever powers are used:

Place value: the ones place is $10^0 = 1$ , so $345 = 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$
Computers: the rightmost digit of a binary number is worth $2^0 = 1$
Growth models: in $P = 500 \times 2^t$ , the starting value at $t = 0$ is $500 \times 2^0 = 500$
Algebra: simplifying $\frac{x^4}{x^4}$ or $3x^0$ needs the rule every time

a^0

were

0

, the ones place would be worth nothing and every growth model would start at zero. The value

1

is the only one that keeps all the exponent rules working.

Real World Applications

Place Value and Powers of Ten

Every place in our number system is a power of ten. The ones place is $10^0 = 1$.

Example:

$6072 = 6 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$ , and the last term is $2 \times 1 = 2$ .

1Try It Yourself

A teacher writes $6072$ in expanded form: $6 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$ .

What is the value of the last term, $2 \times 10^0$ ?

Step 1: Write the mathematical expression

Write the last term as a calculation:

Binary Numbers in Computers

Computers store numbers in binary. Each binary place is a power of two, and the rightmost place is $2^0 = 1$.

Example:

Binary $1011$ means $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$ .

2Try It Yourself

A sensor sends the binary number $101$ , which means $1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$ .

What is this number in our usual decimal system?

Step 1: Write the mathematical expression

Write the sum of the place values:

Starting Value of a Growth Model

Biologists model a population that doubles every hour with $P = P_0 \times 2^t$, where $t$ is the time in hours.

Example:

For $P = 200 \times 2^t$ , after 3 hours $P = 200 \times 2^3 = 1600$ , and at the start $P = 200 \times 2^0 = 200$ .

3Try It Yourself

A culture of bacteria grows according to $P = 500 \times 2^t$ , where $t$ is the number of hours since the start.

How many bacteria are there at the start, when $t = 0$ ?

Step 1: Write the mathematical expression

Put $t = 0$ into the formula:

Key Takeaways

1For any nonzero number $a$ : $a^0 = 1$
2Pattern: each step down in the exponent divides by the base, so $2^1 = 2$ becomes $2^0 = 2 \div 2 = 1$
3Quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$ , and a number divided by itself is $1$
4The exponent belongs only to the base right in front of it: $5x^0 = 5$ but $(5x)^0 = 1$
5Brackets matter with negatives: $-3^0 = -1$ but $(-3)^0 = 1$
6 $a^0$ is never $0$ and never $a$ ; $0^0$ is left undefined at this level

Frequently Asked Questions

Because each step down in the exponent divides by the base. From

5^1 = 5

, one more step down gives

5 \div 5 = 1

. The quotient rule says the same thing:

\frac{5^3}{5^3} = 5^0

, and any nonzero number divided by itself is

1

Glossary

Zero exponent: An exponent equal to $0$ . For any nonzero base, $a^0 = 1$ .
Base: The number that is raised to a power. In $(-3)^0$ the base is $-3$ , in $-3^0$ it is $3$ .
Quotient rule: When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ .
Coefficient: The number in front of a variable. In $5x^0$ the coefficient is $5$ , and it is not affected by the exponent.
Undefined: An expression that is given no value. At this level $0^0$ is left undefined.

Formula Card

Zero exponent rule

a^0 = 1 \quad (a \neq 0)

Any nonzero number to the power of zero equals 1

Why it works

\frac{a^n}{a^n} = a^{n-n} = a^0 = 1

The quotient rule and dividing a number by itself give the same result

Coefficient in front

c \cdot x^0 = c \quad (x \neq 0)

Only the variable becomes 1, the coefficient stays

Brackets and negatives

(-a)^0 = 1, \quad -a^0 = -1

With brackets the base is negative, without brackets the minus sign stays outside

Zero Exponent

Definition

Worked Examples

Common Mistakes

Interactive Visual

Exponents as repeated factors

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Place Value and Powers of Ten

Binary Numbers in Computers

Starting Value of a Growth Model

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Zero exponent rule

Why it works

Coefficient in front

Brackets and negatives

More in This Topic

Introduction to Exponents

Laws of Exponents

Square Roots and Cube Roots

Product Rule for Exponents