For a base a and a rational exponent p/q (integers, q ≠ 0):
- a^(p/q) = (a^p)^(1/q) when interpreted over real numbers.
- a^(p/q) = (qth-root(a))^p is an equivalent view for real roots.
- If a < 0 and q is even, a real value does not exist; the calculator can show the principal complex value.
- Principal complex form: (-|a|)^(p/q) = |a|^(p/q)·(cos(πp/q) + i·sin(πp/q)).
- Select the base input type: decimal or fraction.
- Enter the base value (or numerator and denominator).
- Enter exponent numerator p and denominator q.
- Choose “Real only” or enable complex output if needed.
- Set precision, optionally reduce p/q, and show steps.
- Press Calculate. The result appears above the form.
- Use CSV or PDF buttons to export saved history.
| Base | Exponent | Meaning | Expected |
|---|---|---|---|
| 16 | 1/2 | Square root | 4 |
| 27 | 2/3 | (Cube root)^2 | 9 |
| -8 | 1/3 | Real cube root | -2 |
| -16 | 1/2 | Complex principal | 0 + 4i |
No history yet. Run a calculation to save it.
1) What does a^(p/q) mean?
It represents a fractional exponent. It is equivalent to taking the qth root of a and then raising the result to p, when a real root exists.
2) Why should I reduce the exponent fraction?
Reducing p/q avoids ambiguous forms and improves consistency, especially for negative bases. For example, 2/4 becomes 1/2, which clarifies whether a real root is valid.
3) Can this handle negative bases?
Yes. If q is odd, a real result exists. If q is even, the real result is undefined, but you can enable complex output to see the principal complex value.
4) What is “principal complex value”?
It is the standard single complex answer chosen by the principal branch of the complex logarithm. For negative bases, it uses an angle of π·p/q with magnitude |a|^(p/q).
5) Why does 0^(negative) fail?
Because it requires division by zero. A negative exponent means 1 divided by a positive power of the base, and a power of zero in the denominator is undefined.
6) How accurate are the results?
Results use floating-point arithmetic. You can control rounding with the precision selector. For large magnitudes or extreme exponents, small numerical differences are expected.
7) What do the CSV and PDF downloads include?
They export your saved calculation history from this browser session, including timestamp, base, exponent, mode, and the displayed value.