Free · Powers and roots
Real Nth Root with Domain Checks
Find the principal real nth root with an integer degree, including signed odd roots and explicit negative-even exclusions.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
The real fifth root of−32 is−2 because (−2)⁵=−32.
yⁿ=a; y=sign(a)|a|^(1/n), with negative a allowed only for odd n.
Worked example
Enter these known values and leave the other values blank.
- a
- -32
- n
- 5
- Result
- -2
Assumptions and limitations
- The degree n is an integer greater than1 and below10^50 in this reviewed finite-precision model. The calculator exposes the captured forward calculation, not a general multivalued inverse.
- For even n, a must be nonnegative and the result is the nonnegative principal root. For odd n, negative radicands produce negative real roots. Zero has root zero.
- Finite-precision results can require rounding. Reconstructing yⁿ provides an independent check; extreme exponents can exceed numerical resolution.
- Values smaller than fixed-decimal display resolution use scientific notation while retaining the computed finite precision; this avoids silently displaying nonzero values as zero.
Common questions
Can a negative number have a real root?
Yes for an odd degree, such as the cube root of−8=−2. An even-degree root of a negative number is not real.
Does an even root include both signs?
This calculator returns the nonnegative principal root. Solving the equation yⁿ=a may have more than one root.
References
Replace the captured helper parity test on reciprocal exponent with a signed real nth-root expression; even roots of negative numbers have no real result. Preserve the forward-only controls. Use exact degree membership, explicit negative-even real-domain rejection and a degree below10^50 for reviewed finite-precision parity. Keep the visible no-real-root outcome and every degree/radicand control. Preserve finite nonzero results in scientific notation only when the existing 1000-decimal rounding search cannot meet its own relative precision; ordinary formatting is unchanged.