Zero and negative exponents is one of 3 exponents formulas in the algebra section of this library, and it is used at middle school · high school level.
Why zero and negative exponents works
Both definitions are forced by extending the subtraction rule downward. Dividing a cubed by a cubed gives a to the zero by subtracting, and it obviously equals 1, so a to the zero must be 1. The same argument on a squared over a to the fifth turns a to the -3 into 1 over a cubed.
What each symbol means
$a$ is the nonzero base and $n$ is a positive integer.
Zero and negative exponents: when it holds
$a\ne0$; $0^0$ is not assigned the value $1$ by this rule.
When it stops applying
Both rules need a nonzero base. Zero to the -2 asks for 1 over 0, which does not exist, and zero to the zero is left undefined because different ways of approaching it give different answers.
Zero and negative exponents: a worked example
$2^{-3}=1/2^3=1/8$.
The mistake to avoid
What people do: Reading 2 to the -3 as -8, as if the minus sign belonged to the answer.
Why it goes wrong: A negative exponent flips the value into a fraction; it never makes the value negative. The real answer is 1 over 8, which is 0.125, a positive number.
Do this instead: Say it as a reciprocal instruction: the minus sign means move the power to the other floor of the fraction, then apply it as usual.
Zero and negative exponents: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is the nonzero base and $n$ is a positive integer.
- Check the conditions before substituting. $a\ne0$; $0^0$ is not assigned the value $1$ by this rule.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Exponents
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where zero and negative exponents comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about zero and negative exponents
Is a negative base with a negative exponent negative?
It can be, but for the sign reason, not the exponent reason. Since -2 cubed is -8, the value of -2 to the -3 is -1 over 8, so the minus survives from the odd power.
What does an exponent of -1 mean?
It means the reciprocal, so x to the -1 is 1 over x. This is why a fraction raised to the -1 simply turns upside down.
How do I simplify a negative exponent that is already in the denominator?
Move it up to the top and drop the minus sign. So 1 over x to the -3 is x cubed, because two reciprocals in a row undo each other.
Why is zero to the zero left undefined?
Because the two patterns that justify the rules disagree there. Any nonzero number to the zero is 1, but zero to any positive power is 0, and nothing settles the tie at the corner.