Nth Root Calculator

Calculate an nth root with support for positive radicands and negative radicands when the root index is an odd integer

Negative numbers require an odd integer root index for a real-valued result.

Your result will appear here.

About the Nth Root Calculator

Nth Root Calculator generalizes square and cube roots by allowing any positive integer root degree. It answers the question: what value raised to the nth power gives the number entered?

Whether a real result exists depends on both the input and the root degree. Negative numbers have real odd roots, but even roots of negative numbers are not real.

How to Use the Nth Root Calculator

  1. Enter the value whose root you want.
  2. Enter the positive integer root degree n.
  3. Calculate the real result when one exists.
  4. Check whether the chosen degree is even or odd before entering a negative radicand.

How the Calculation Works

The nth root of x is x^(1/n), meaning a value y such that y^n = x.

For positive x, a principal positive real root exists for any positive integer n. For negative x, a real root exists only when n is odd in the ordinary real-number system.

Worked example

Nth Root Calculator Worked Example

Fifth root of 32

The fifth root of 32 is 2 because 2⁵ = 32. The fifth root of −32 is −2 because (−2)⁵ = −32.

Result guide

Understanding Your Nth Root Calculator Result

The result is the inverse of an nth power. Large root degrees move positive values toward 1, while the domain rules determine whether negative inputs remain real.

Before you rely on the answer

Assumptions and Limitations

  • Root degree is a positive integer.
  • Real-number mode is being used.
  • Displayed decimals are approximations when the root is irrational.

Practical tip

Using This Result in Practice

When checking a root manually, raise the result back to the original degree rather than multiplying by the degree. Root and division operations are not interchangeable.

Related calculations

Useful Calculations to Continue With

For the common second-root case, use the Square Root Calculator.

For the common third-root case, use the Cube Root Calculator.

Common questions

Nth Root Calculator FAQ

What does nth root mean?

It is the value that must be raised to power n to reproduce the original number.

Can an even root of a negative number be real?

No. In the real-number system, even powers are non-negative, so an even root of a negative input has no real result.

Can an odd root of a negative number be real?

Yes. For example, the cube root of −8 is −2.