Fractional Exponent Calculator

A fractional exponent $$\frac{p}{q}$$ decomposes into two operations:

$$a^{p/q} = \sqrt[q]{a^p} = (\sqrt[q]{a})^p$$

Method 1 — Power then root: First compute $$a^p$$, then take the $$q$$-th root: $$\sqrt[q]{a^p}$$.

Method 2 — Root then power: First compute $$\sqrt[q]{a}$$, then raise to the $$p$$-th power: $$(\sqrt[q]{a})^p$$.

Both methods yield the same result. The calculator uses $$a^{p/q} = e^{(p/q) \cdot \ln(a)}$$ internally for maximum precision, and displays both decompositions for educational clarity.

When $$p$$ is negative, the result is a reciprocal: $$a^{-p/q} = \frac{1}{a^{p/q}}$$.

The Result shows $$a^{p/q}$$ computed directly. The Exponent as Decimal shows the fractional exponent in decimal form. base^p shows the intermediate step of raising to the full integer power. The qth Root and Root First fields demonstrate both decomposition paths, confirming they yield identical results (within floating-point precision).