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Math & Statisticsalgebra

Cube Root and Nth Root Calculator

Any root of a number, computed as a fractional power. Odd roots of negative numbers are real and are handled; even roots of negatives are not.

Also called: cube root calculator, radical calculator.

Root
4

The 5 root of 1,024 is 4. Raised back to the power 5 it gives 1,024.

Raised back, as a check
1,024
As a fractional power
1024 ^ (1/5)
Worth knowing
An odd root keeps the sign of the number, so negatives are fine.
How this is calculated
Method and background

How this is calculated

The k-th root is the number that, raised to the power k, returns the original. That is the same as raising to the power one over k, which is how it is computed. The asymmetry worth knowing: an odd root of a negative number is real, because a negative multiplied an odd number of times stays negative, while an even root of a negative has no real value at all.

root = value ^ (1 / k)
n
The number
k
Which root

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

an odd root of a negative is real

Number
-27
Root
3

Root-3

boundary: the case that separates odd roots from even

Open this example

Method and limits

What it assumes

  • Real numbers only, and the principal root where several exist.

What it deliberately does not model

  • Even roots of negative numbers have no real value and are declined.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why can I take the cube root of a negative number but not the square root?
Because multiplying a negative by itself an odd number of times keeps the sign. Minus two cubed is minus eight, so the cube root of minus eight is minus two. Squaring always produces a positive, so no real number squares to a negative.