Fractional exponents combine powers and roots: the denominator is a root and the numerator a power. 8^(2/3) = (∛8)² = 4. They appear in scaling laws, compound growth and physics formulas.
How to use the Fraction Exponent Calculator
- Solve for
- Base x
- Numerator n
- Denominator d
- Result a
Results update as you type, and everything is calculated in your browser.
How it works
Choose the unknown. The base is found by raising a to d/n; the exponent parts by logarithms.
Inputs explained
| Field | What to enter |
|---|---|
| Solve for | Choose one: Base x, Numerator n, Denominator d, Result a |
| Base x | A number |
| Numerator n | A number |
| Denominator d | A number |
| Result a | A number |
Example values
How the main result changes when Base x changes and everything else stays at the example values above:
| Base x | Unknown value |
|---|---|
| 4 | 2.5198 |
| 6 | 3.3019 |
| 8 | 4 |
| 10 | 4.6416 |
| 12 | 5.2415 |
| 16 | 6.3496 |
Formula
x^(n/d) = (ᵈ√x)ⁿ = ᵈ√(xⁿ)
x^(−n/d) = 1 / x^(n/d)
Worked example
Inputs — Solve for: Result a; Base x: 8; Numerator n: 2; Denominator d: 3.
Results — Unknown value = 4; Exponent n/d = 0.6666666667.
Key terms
- Fractional exponent
- An exponent of the form n/d, meaning the d-th root of the n-th power.
Frequently asked questions
What is 8^(2/3)?
(∛8)² = 2² = 4.
What does a negative fractional exponent mean?
A reciprocal: 16^(−1/2) = 1/4.
Is x^(1/2) the same as √x?
Yes.
Can the base be negative?
Only when the denominator is odd, e.g. (−8)^(1/3) = −2.