Type the base in “Rational base” and the exponent in “Integer exponent”; when you select “Calculate,” the result appears already in lowest terms. The example below combines a negative base with a negative exponent. For the theory, including fractional exponents, see the guide to negative and fractional exponents; this page calculates with integer exponents.
Worked example: (−2/3) to the power −3
Calculate (−2/3)^−3 and decide whether the result is positive or negative.
- Flip the base. The exponent is negative, so take the reciprocal and make the exponent positive: (−2/3)^−3 = (−3/2)^3. The sign stays with the fraction.
- Decide the sign. The exponent 3 is odd, so the result is negative.
- Raise the numerator and the denominator. 3³ = 27 and 2³ = 8, so (−3/2)^3 = −27/8.
- Write it as a mixed number if you need to. 27 ÷ 8 is 3 with a remainder of 3, so −27/8 = −3 3/8.
Answer: (−2/3)^−3 = −27/8.
Check: (−2/3)^3 = −8/27, and the product of −8/27 and −27/8 is 1. They are reciprocals, as two powers with opposite exponents must be.
How to raise a fraction to a power
Raising to the power n means multiplying the fraction by itself n times, which is the same as raising the numerator and the denominator separately: (a/b)^n = a^n/b^n. With 7/10 squared, 7² = 49 and 10² = 100, so the answer is 49/100. A common mistake is to raise only the numerator or to multiply the fraction by the exponent, which would give 14/10.
The negative exponent comes from following that rule downward. Each time the exponent drops by one, the value is divided by the base. From (7/10)^1 to (7/10)^0 you divide 7/10 by 7/10 and get 1, and one more step gives 10/7. That is why (a/b)^−n = (b/a)^n when a is not zero: a negative exponent asks for the reciprocal, not the opposite, and the result, 100/49, is still positive. The same ladder explains why the exponent 0 gives 1 for any nonzero base.
A decimal or a mixed number is converted to a fraction first: 0.25 is 1/4, so 0.25 squared is 1/16, and 1 1/2 is 3/2, so its square is 9/4.
What sign and form the result has
The sign depends on the base and on whether the exponent is even or odd, and the form depends on whether the exponent is negative. The table gathers the five cases worth telling apart.
| Case | Example | What happens |
|---|---|---|
| Negative base, even exponent | (−1/3)^2 = 1/9 | The result is positive |
| Negative base, odd exponent | (−1/3)^3 = −1/27 | The result is negative |
| Positive base, negative exponent | (5/2)^−2 = 4/25 | It flips and stays positive |
| Negative base, negative exponent | (−1/5)^−3 = −125 | It flips and the sign follows the parity |
| Zero exponent | (9/8)^0 = 1 | Gives 1 for any nonzero base |
With an even exponent the sign disappears and with an odd one it stays, and a negative exponent only swaps the fraction for its reciprocal. If the base is between 0 and 1, its power with a positive exponent (2 or more) is smaller than the base. If the base is greater than 1, the power with a negative exponent is smaller than the base.
Practice the three cases
The three exercises cover the even exponent, the odd exponent and the zero exponent, which are the cases where mistakes happen most. Solve them by hand and then check with the calculator.
Practice
- Calculate (−4/5)^2, with an even exponent.
- Calculate (−2/5)^3, with an odd exponent.
- Calculate (3/10)^0, with a zero exponent.
Show the answers
- 16/25, because 4² = 16, 5² = 25 and the sign disappears.
- −8/125, because 2³ = 8, 5³ = 125 and the sign stays.
- 1, because the base is not zero.
What to do when the exponent is not an integer
This calculator accepts only integer exponents from −100 to 100, and it rejects 0^0 and zero with a negative exponent. For an exponent such as 1/2, which is the same as a root, use fraction roots. If the power sits inside an expression with sums and parentheses, the fraction expression calculator follows the order of operations. The guide to negative and fractional exponents explains the theory, and the guide to the reciprocal of a fraction covers the reciprocal on its own. It is the same reciprocal used to divide fractions.
The English Wikipedia article Exponentiation gives the identity b^−n = 1/b^n for nonzero b, defines a nonzero number to the power 0 as 1, and notes that 0^0 is often left undefined, for example in analysis, though some fields such as combinatorics set it to 1 (accessed October 17, 2026). This calculator treats 0^0 as undefined.
Frequently asked questions
Why doesn't the calculator solve 0^0?
Because it has no single value: a nonzero number to the power 0 is 1, but 0 to a positive power is 0, and the two rules clash. The calculator marks it as undefined instead of substituting 1. It also does not accept 0 with a negative exponent, such as 0^−2, because that would require dividing by zero. With a positive exponent, 0^4 = 0.
How do I multiply powers of the same fraction?
Keep the base and add the exponents: (1/3)^2 × (1/3)^3 = (1/3)^5 = 1/243. It works the same with exponents of different signs: (4/5)^6 × (4/5)^−2 = (4/5)^4 = 256/625. To check, calculate each power separately and multiply the results.
What happens with a very large exponent?
The calculator does not round: (3/2)^100 is 3^100/2^100, with a numerator of 48 digits and a denominator of 31, both shown in full. If you need an approximate value, convert the result to a decimal with the fraction to decimal calculator.