A fractional exponent m/n means “take the root of index n, then the power m,” and a negative exponent means “flip the base.” With a fraction, both rules apply to the numerator and the denominator. So (8/27)^(2/3) = 4/9, and the result is an exact fraction whenever the root is exact.
This guide explains the theory. To compute with whole-number exponents, including negative ones, use the fraction powers calculator. For roots, the fraction roots calculator tells you when the result is not a fraction.
| Exponent | What it means | Example |
|---|---|---|
| Negative, −n | Flip the base and use n | (3/4)⁻² = (4/3)² = 16/9 |
| Zero | Gives 1 if the base is not 0 | (5/7)⁰ = 1 |
| Fractional, 1/n | Root of index n | (25/16)^(1/2) = 5/4 |
| Fractional, m/n | Root of index n, then power m | (81/16)^(3/4) = (3/2)³ = 27/8 |
| Negative and fractional | Flip, then root and power | (4/9)^(−1/2) = (9/4)^(1/2) = 3/2 |
To decide, look at the sign of the exponent first and then at its denominator, which is the index of the root. The last row combines the two rules: you can flip first or take the root first, because with a positive base the result is the same. Flipping first avoids working with negative exponents.
Worked example: (8/27)^(2/3)
Compute this power and write the result as a fraction.
- Check that the exponent is in lowest terms. It is: 3 is the index of the root and 2 is the power.
- Take the cube root. ∛8 = 2 and ∛27 = 3, so the root of the base is 2/3.
- Square it. 2² = 4 and 3² = 9, so the result is 4/9.
Result: (8/27)^(2/3) = 4/9.
Check: cubing 4/9 gives 64/729, and squaring the base gives the same, because 8² = 64 and 27² = 729. Both routes agree.
How to compute a power with a fractional exponent
Write the exponent as a fraction in lowest terms. If it is 4/6, reduce it before you start, because the index of the root is the reduced denominator. Then take the root of the numerator and the denominator of the base separately, and raise the result to the power given by the numerator of the exponent.
Starting with the root keeps the numbers small. In the example above, the reverse route forces you to compute 8² = 64 and 27² = 729 and then take the cube root of 64/729. You get the same answer with more work.
The result is an exact fraction when, in the base already simplified, the numerator and the denominator are perfect powers of the index. That is why (50/18)^(1/2) is exact: 50/18 reduces to 25/9 and √(25/9) = 5/3. You can reduce with the simplify fractions tool before looking for the root.
Why a negative exponent is a reciprocal
Powers of the same base are divided by subtracting exponents, and both definitions come from that. With a nonzero base a, aⁿ ÷ aⁿ = 1 and also aⁿ ÷ aⁿ = a⁰, so a⁰ = 1. If you now divide a⁰ by aⁿ, you get a⁻ⁿ = 1/aⁿ. A negative exponent is therefore the multiplicative inverse of the positive power.
With fractions, that inverse is the one you already know: swap the numerator and the denominator. So (2/5)⁻³ = (5/2)³ = 125/8. The guide to the reciprocal of a fraction covers that step with whole numbers, mixed numbers and negatives. It is the same reciprocal the divide fractions tool uses when it turns a division into a product.
What happens with a negative base
With a negative base, the root may not exist. If the index is odd, the negative root exists and the sign is kept: (−8)^(1/3) = −2, because (−2)³ = −8. If the index is even, no real number raised to an even power gives a negative result, and the square root of −4 is not real.
To find the index, reduce the exponent first. In (−8)^(2/6) the reduced exponent is 1/3, with an odd index, and the value is −2. If you do not reduce 2/6, some programs read it as 64^(1/6) = 2, which does not match −2. That is why you should always reduce the exponent before deciding on the index.
The order of operations also matters. In −2² the power is computed before the sign and the value is −4; in (−2)² the base is −2 and the value is 4. The same happens with fractions: −(1/5)² = −1/25, while (−1/5)² = 1/25.
When the result is not a fraction
If the root is not exact, the result is an irrational number, and no fraction of whole numbers equals it. √2 ≈ 1.4142 (error less than 0.0001). The exact form is the radical.
A fractional base works the same way: (1/5)^(1/2) = 1/√5 = √5/5 ≈ 0.4472 (error less than 0.0001). To write it without a radical in the denominator, the guide to rationalizing denominators explains the step from 1/√5 to √5/5.
When the base has letters instead of numbers, you need to add conditions on the values that make the denominator zero or the radicand negative. That is covered in algebraic fractions and domain. If you want to combine powers with sums and products on one line, the fraction expressions tool respects the order of operations.
Practice
- Compute (2/5)⁻³.
- Compute (16/81)^(1/4).
- Compute (9/25)^(−3/2).
Show the answers
- 125/8, because (2/5)⁻³ = (5/2)³.
- 2/3, because 2⁴ = 16 and 3⁴ = 81.
- 125/27: flip to (25/9)^(3/2), the square root is 5/3 and its cube is 125/27.
Sources
The English Wikipedia article “Exponentiation” covers the extension of powers to negative integer exponents, the value 1 for a zero exponent with a nonzero base, the use of fractional exponents to write roots, and the existence problems with negative bases (accessed October 17, 2026).