Fractional Exponent Calculator — Rational Powers
Compute any rational power x^(p/q) — equal to the q-th root of x raised to the p-th power. Enter the base and the numerator and denominator of the exponent to get the exact numeric result and a plot of the function.
Base raised to the fractional power p/q
- 1
Fractional exponent p ÷ q
2 ÷ 3 = 0.666667 - 2
Raise base to the exponent
8 to the power 0.666667 = 4x^(p/q) equals the q-th root of x raised to the p-th power.
How does this calculator work?
x^(p/q) equals the q-th root of x raised to the p-th power. Enter the base and the exponent fraction (p over q) — the result follows from the power-of-a-power rule. For negative bases, a real answer exists only when q is an odd integer.
Formula
How this is calculated
A fractional exponent x^(p/q) is shorthand for raising x to the power p/q. By the power-of-a-power rule (x^a)^b = x^(a·b), this is the same as taking the q-th root of x first, then raising to the p-th power: (ᵠ√x)^p. The order can also be reversed — raise x to p first, then take the q-th root — because both give the same result when x ≥ 0.
The result is real for any x ≥ 0 and any real p/q. For negative x the result is real only when q is an odd integer — that is, the q-th root of a negative number is itself real (e.g. ∛(−8) = −2). Even denominators would require complex numbers and so this calculator returns no result for those cases.
The plot shows y = t^(p/q) for t ≥ 0 around your base, giving a visual sense of how the function behaves. A numerator smaller than the denominator produces a curve that grows slowly (concave); one larger than the denominator grows quickly (convex). When p/q < 0 the function approaches infinity near zero and decays toward zero for large t.
Frequently asked questions
x^(2/3) means take the cube root of x and then square the result: (∛x)². For x = 8 this gives (∛8)² = 2² = 4. The denominator is the root order and the numerator is the power applied after rooting.
Only when the denominator q is an odd integer (3, 5, 7 …). For example (−8)^(1/3) = −2 is valid, but (−4)^(1/2) is not real. Even-index roots of negative numbers are complex, which this calculator skips.
They are the same operation: x^(p/q) = ᵠ√(xᵖ). The denominator of the fraction is the index of the radical (2 = square root, 3 = cube root, etc.) and the numerator is the power inside the radical.
Also known as
TG we-Calculate Editorial Team. (2026). Fractional Exponent Calculator — Rational Powers [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/fractional-exponent-calculator
TG we-Calculate Editorial Team. "Fractional Exponent Calculator — Rational Powers." TG we-Calculate. 2026. https://we-calculate.com/calculator/fractional-exponent-calculator.
TG we-Calculate Editorial Team, "Fractional Exponent Calculator — Rational Powers," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/fractional-exponent-calculator
@misc{wecalculate_fractional_exponent_calculator, title = {Fractional Exponent Calculator — Rational Powers}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/fractional-exponent-calculator}}, year = {2026}, note = {TG we-Calculate} }
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