Exponent Calculator · Practical guide
How to Calculate Negative Exponents
A negative exponent means take the reciprocal: x^-n = 1/x^n, so 2^-3 = 1/8 = 0.125.
At a glance
The One Rule You Need
A negative exponent does not make the answer negative — it makes it a reciprocal. The rule is x^-n = 1/x^n. So 2^-3 = 1/2^3 = 1/8 = 0.125, and 10^-2 = 1/100 = 0.01. To evaluate any negative exponent, flip the base to the denominator and make the exponent positive.
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Enter your own values, then compare the result with the examples below.
Results
Worked Examples
| Expression | Reciprocal Form | Decimal Result |
|---|---|---|
| 2^-1 | 1/2 | 0.5 |
| 2^-2 | 1/4 | 0.25 |
| 2^-3 | 1/8 | 0.125 |
| 3^-2 | 1/9 | 0.1111 |
| 4^-2 | 1/16 | 0.0625 |
| 10^-3 | 1/1,000 | 0.001 |
Negative Fractional Exponents
When a negative exponent is also a fraction, apply both rules: the denominator means a root, the negative sign means a reciprocal. For example, 8^(-1/3) = 1 / 8^(1/3) = 1/2 = 0.5, because the cube root of 8 is 2. Similarly, 4^(-3/2) = 1 / (sqrt(4))^3 = 1/8 = 0.125.
Common Mistakes to Avoid
- 2^-3 is 0.125, not -8. The sign of the exponent never changes the sign of the result.
- (-2)^-3 = 1/(-2)^3 = -0.125 — here the negative is inside the base, so it survives.
- Anything to the power of 0 is 1, including 5^0 and (-7)^0.
- 1^-100 = 1; the reciprocal of 1 is still 1.
Enter any base and a negative exponent above to see the result along with the formula and explanation.
Frequently Asked Questions
Is a negative exponent the same as a negative number?
No. A negative exponent produces a positive fraction when the base is positive: 2^-3 = 0.125. The result is only negative if the base itself is negative and the exponent is an odd integer.
What is 0 to a negative exponent?
It is undefined. 0^-1 would be 1/0, which is division by zero, so negative exponents with a base of zero have no value.
How do negative exponents appear in real life?
Scientific notation uses them constantly: a micrometer is 10^-6 meters and a nanogram is 10^-9 grams. Decay formulas in physics and finance also use negative exponents.
How do I move a negative exponent across a fraction?
A factor with a negative exponent flips to the other side of the fraction bar and becomes positive: x^-2 / y^-3 = y^3 / x^2.