Math
Negative Exponent Calculator
A negative exponent means take the reciprocal: a^-n = 1 / a^n.
Enter any base (a number, a fraction, or a letter) and a negative exponent to see the rewritten positive-exponent form, the flipped fraction when the base is a fraction, and the exact value.
A number (2, -3, 0.5), a fraction (2/3), or a letter (x).
Usually negative here, but any exponent works.
Try an example
Result
Rewritten
—
- Value
- —
- Original
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
The negative exponent rule
The small raised number saying how many times to multiply the base by itself. count repeated multiplication, and each step down divides by the The number being raised to a power, or the side a shape's height is measured from.: 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1. Keep dividing and you get 2^-1 = 1/2, 2^-2 = 1/4, 2^-3 = 1/8. Negative exponents continue the pattern into fractions.
Pattern table for base 2
| Power | Value |
|---|---|
| 2^3 | 8 |
| 2^2 | 4 |
| 2^1 | 2 |
| 2^0 | 1 |
| 2^-1 | 1/2 = 0.5 |
| 2^-2 | 1/4 = 0.25 |
| 2^-3 | 1/8 = 0.125 |
Fraction bases flip
Negative exponents in the denominator
A negative exponent underneath a fraction bar moves up: 1 / x^-2 = x^2. In general, any factor can cross the fraction bar by changing the sign of its exponent.
How to use it
- Enter the base. Fractions like 2/3 and negatives like -3 are fine.
- Enter the negative exponent.
- Read the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. form, the flipped fraction when the base is a fraction, and the evaluated value.
How to read the answer
A negative exponent never makes the answer negative. It moves the power to the other side of a fraction bar: 2^-3 is 1/8, a small positive number. Only a negative base can make the result negative, and only when the exponent is odd.
Common mistakes and edge cases
- Thinking 2^-3 = -8. It is 1/8.
- Forgetting to flip a fraction base: (2/3)^-2 = (3/2)^2 = 9/4.
- Leaving the negative sign on the base when flipping: (-2)^-3 = 1/(-2)^3 = -1/8.
- Applying the exponent to the The number multiplying a variable, like the 3 in 3x.: 3x^-2 = 3 / x^2, not 1 / (3x^2).
Worked examples
Basic
2^-3 = 1/8
Rewritten
1 / 2^3
Fraction base flips
(2/3)^-2 = (3/2)^2 = 9/4
Rewritten
(3/2)^2
Negative base, odd exponent
(-2)^-3 = -1/8
Rewritten
1 / (-2)^3
Variable base
x^-2 = 1/x^2
Rewritten
1 / x^2
Decimal base
0.5^-2 = 4
Rewritten
2^2
Positive exponent (no change)
5^2 stays 5^2
Rewritten
5^2
Undefined
0^-1 divides by zero
Rewritten
Error
Frequently asked questions
What does a negative exponent mean?+
Take the reciprocal of the positive power: a^-n = 1 / a^n. So 2^-3 = 1/2^3 = 1/8.
Is 2 to the negative 3 equal to -8?+
No. 2^-3 = 1/8 = 0.125. A negative exponent makes the value small, not negative.
How do you simplify a fraction with a negative exponent?+
Flip the fraction and make the exponent positive: (2/3)^-2 = (3/2)^2 = 9/4.
What is x to the negative 2?+
x^-2 = 1 / x^2.
Can a negative exponent give a negative answer?+
Only if the base is negative and the exponent is odd: (-2)^-3 = -1/8. Otherwise the answer is positive.
Why is 0 to a negative power undefined?+
Because 0^-n = 1 / 0^n = 1 / 0, and division by zero has no value.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Exponent Calculator
Calculate powers with positive, negative, and decimal exponents while showing the exponent rule used.
Multiplying Exponents Calculator
Multiply powers with the same base (add exponents) or the same exponent (multiply bases), with the rule named, symbolic or numeric bases, and the exact evaluated value.
Dividing Exponents Calculator
Divide powers with the same base (subtract exponents) or the same exponent (divide bases), with the quotient rule named, negative results rewritten as reciprocals, and exact values.
Power of a Power Calculator
Simplify (a^m)^n by multiplying the exponents, with numeric or variable bases, negative and fractional exponents, and the exact evaluated value shown with steps.
Fraction Calculator
Add, subtract, multiply, divide, simplify, compare, or convert decimals to fractions with steps and visual fraction bars.
Fractional Exponent Calculator
Evaluate a base raised to a fractional (rational) exponent such as 8^(2/3) exactly when possible, with the root-then-power steps, simplest radical form, decimal value, and negative-exponent handling.
Last updated: September 4, 2026