1
Roots and powers combined
Base, numerator, denominator in — exact or decimal answer out.
MATH TOOL
Fractional Exponent Calculator
A fractional exponent like 272/3 is just a root and a power in disguise: take the b-th root, then raise to the a. Enter all three numbers and see every step.
The rule: xa/b = (b-th root of x)a
A fractional exponent packs two operations into one. The denominator is a root, the numerator is a power — do the root first, then the power (or the power first, then the root; the answer is the same). So 272/3 means "cube-root 27, then square it": ∛27 = 3, and 3² = 9. Doing the root first keeps the numbers small, which is why the calculator shows the steps in that order.
A worked example
Evaluate 163/4. Denominator 4 says fourth root: ⁴√16 = 2 (because 2&sup4; = 16). Numerator 3 says cube it: 2³ = 8. Check by the other order: 16³ = 4,096, and ⁴√4,096 = 8 — same answer, much uglier arithmetic. Root-first is the civilized path. Another: 813/4 = (⁴√81)³ = 3³ = 27.
Output · formula · meaning
| Output | Formula | What it means |
|---|---|---|
| Result | xa/b | The value — 272/3 = 9. |
| Root step | ʸ-th root of x | Denominator first: ∛27 = 3. |
| Power step | roota | Then the numerator: 3² = 9. |
| Exactness | integer check | Exact when the root comes out whole — 9, not 8.9999999. |
Who uses this
- Students: fractional exponents are an algebra staple — the root-then-power reading turns a scary expression into two familiar steps.
- Science students: growth laws, scaling laws, and half-life formulas all hide fractional powers (square roots are just x1/2).
- Programmers: game physics and easing curves use fractional powers constantly — knowing 163/4 = 8 by hand keeps the intuition sharp.
- Finance learners: the CAGR formula's (end/start)1/years is a fractional exponent — this is the engine inside compound growth.
Accuracy & limitations
The calculator is exact whenever the root is a whole number — 272/3 comes back as clean 9, not 8.9999999997. Non-whole roots are computed by logarithms and rounded for display, which is accurate to about 15 significant digits. Two hard limits, both mathematical: the denominator can't be zero (a "zeroth root" means nothing), and a negative base with an even denominator has no real answer — (−16)1/2 is imaginary, and the calculator says so instead of guessing.
For plain squares, cubes, and roots, the squares, cubes & roots chart covers 1–30 in a printable table.
Cite this calculatorBright Side Kit. “Fractional Exponent Calculator.” brightoolbox.com, 2026.
Bright Side Kit. "Fractional Exponent Calculator." https://brightoolbox.com/fractional-exponent/ Accessed October 22, 2026.Frequently asked questions
What is 27 to the power of 2/3?
9. Take the cube root first (denominator): ∛27 = 3, then square it (numerator): 3² = 9. Root-first keeps the arithmetic tiny.
What does a fractional exponent mean?
Two operations in one: the denominator is a root and the numerator is a power. xa/b = (the b-th root of x), raised to the a. So 163/4 = (⁴√16)³ = 2³ = 8.
How do you solve 16^(3/4) by hand?
Read the denominator as the root: the 4th root of 16 is 2, since 2 × 2 × 2 × 2 = 16. Read the numerator as the power: 2³ = 8. Answer: 8.
Can you do the power before the root?
Yes — (xa)1/b always equals (x1/b)a. But power-first explodes the numbers: 16³ = 4,096 before the root, versus 2 before the power. Root-first is the sane order.
What is 8^(2/3)?
4. Cube root of 8 is 2, and 2² = 4.
Can a fractional exponent have a negative base?
Sometimes. (−27)2/3 = 9 is fine — the cube root of −27 is −3, and (−3)² = 9. But (−16)1/2 asks for the square root of a negative number, which has no real answer — the calculator flags it instead of returning nonsense.
What is x^(1/2)?
The square root of x. x1/3 is the cube root. Fractional exponents with numerator 1 are just roots in disguise — that’s why the denominator is the root.