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Math / Root

Root Calculator

Find the principal nth root of a real number, including odd roots of negative values. The index must be a positive integer, and even roots of negative radicands are rejected because they are not real. The nth root r satisfies rⁿ=x. For even n, x must be nonnegative; for odd n, the real root keeps the sign of x. For an even index there are two signed solutions to rⁿ=x when x is positive, but the principal root returned here is the nonnegative one.

Your inputs

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Calculator settings

Your result
-5
Principal real root

Example inputs shown. Adjust them to explore.

A familiar situation

Find the cube root of a signed volume or another real number with a chosen index.

How to use this calculator

  1. Enter radicand, root index.
  2. Select Calculate to see the answer and its breakdown.
  3. Keep a scenario, change an input and compare the results.

Use consistent units and check the assumptions below before applying the result.

How root is calculated

The nth root r satisfies rⁿ=x. For even n, x must be nonnegative; for odd n, the real root keeps the sign of x. For an even index there are two signed solutions to rⁿ=x when x is positive, but the principal root returned here is the nonnegative one. The cube root of −125 is −5 because (−5)³=−125.

rⁿ=x; the principal real root is used.
A worked example

The cube root of −125 is −5 because (−5)³=−125.

Assumptions & useful context

The result depends on the values you enter and the method described above. Displayed numbers may be rounded; calculations use the unrounded values. Check that this model matches your situation before relying on it.

Frequently asked questions

What does principal root mean for an even index?

It is the nonnegative root.

Can the root index be zero?

No; a zero index does not define a root operation.

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