Rational Exponents Calculator — x^(p/q) Fractional Powers
Enter a base x and a fractional exponent p/q to compute x^(p/q). The calculator breaks the calculation into a q-th root step followed by a p-th power step, shows both intermediate values, and plots the function y = x^(p/q) so you can see how your result fits the curve.
8 ^ (2 / 3)
Identify the fractional exponent
Take the q-th root of x (x^(1/q))
Raise the root to the p-th power
Result
- 1
Fractional exponent p ÷ q
2 ÷ 3 = 0.666667 - 2
q-th root of x (x^(1/q))
8^(1 ÷ 3) = 2 - 3
Raise root to power p
2^2 = 4Equals x^(p/q) by the root-then-power rule.
How does this calculator work?
x^(p/q) means take the q-th root of x, then raise to the power p. Example: 8^(2/3) = (∛8)² = 2² = 4. The denominator q defines the root; the numerator p defines the power applied after. Even denominators require a non-negative base for a real result.
Formula
How this is calculated
A rational exponent is a fraction p/q used as a power. The denominator q indicates a root — x^(1/q) is the q-th root of x — and the numerator p indicates a further power. So x^(p/q) = (x^(1/q))^p: first take the q-th root of x, then raise that result to the p-th power. Equivalently you can raise first and root second, x^(p/q) = (x^p)^(1/q), which gives the same answer for all inputs where both paths are defined. The step-by-step view shows the root-first approach.
A rational exponent can also be evaluated directly by converting p/q to a decimal exponent and applying the standard power function. All three routes — root-then-power, power-then-root, and decimal exponent — agree whenever the expression is defined over the real numbers. Most calculators use the decimal route internally because it handles all cases uniformly.
The main restriction is a negative base with an even denominator: (−4)^(1/2) requires a square root of a negative number, which produces a complex (imaginary) result. This calculator returns no result for that case and shows a warning. A negative base with an odd denominator is fine because the odd root of a negative number is negative: (−8)^(1/3) = −2.
Frequently asked questions
x^(1/2) is the square root of x. More generally, x^(1/n) is the n-th root of x. Rational exponents unify root notation and power notation into a single form.
Yes — both paths give the same answer of 4. ∛8 = 2, then 2² = 4. Or 8² = 64, then ∛64 = 4. Either order works when the base is non-negative.
The even root (square root, fourth root, etc.) of a negative number is not real. For example (−4)^(1/2) = 2i. This calculator works only with real numbers, so such cases are flagged with a warning.
Also known as
TG we-Calculate Editorial Team. (2026). Rational Exponents Calculator — x^(p/q) Fractional Powers [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rational-exponents-calculator
TG we-Calculate Editorial Team. "Rational Exponents Calculator — x^(p/q) Fractional Powers." TG we-Calculate. 2026. https://we-calculate.com/calculator/rational-exponents-calculator.
TG we-Calculate Editorial Team, "Rational Exponents Calculator — x^(p/q) Fractional Powers," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rational-exponents-calculator
@misc{wecalculate_rational_exponents_calculator, title = {Rational Exponents Calculator — x^(p/q) Fractional Powers}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rational-exponents-calculator}}, year = {2026}, note = {TG we-Calculate} }
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