Nth Root Calculator
Any Root, One Formula
Square roots and cube roots are really just special cases of a more general idea: the nth root, which asks what number, raised to the nth power, produces the value entered. This calculator generalizes the pattern to any index n, handling the extra wrinkle that comes with negative bases — a real root only exists there when n is an odd whole number.
The Formula
ⁿ√x = x^(1/n)
When x is negative and n is an odd integer (such as 3, 5, or 7), a real result exists and is computed as the negative of the nth root of |x|. When x is negative and n is even or non-integer, no real root exists, and the calculator returns an error rather than a misleading value.
Where Nth Roots Are Used
- Finance — computing a compound annual growth rate (CAGR) requires taking the nth root of a total growth ratio.
- Geometry and scaling — finding a linear dimension from an n-dimensional volume or hypervolume.
- Statistics — the geometric mean of n values is the nth root of their product.
- Engineering — solving for a base variable in formulas where it appears raised to an arbitrary power.
| Number | Root Degree (n) | Result |
|---|---|---|
| 81 | 4 | 3 |
| 32 | 5 | 2 |
| −8 | 3 | −2 |
How to Use This Calculator
- Enter the Number.
- Enter the Root Degree (n).
- Select Calculate to see the nth root, rounded to six decimal places.
Related Calculations
For the two most common special cases, see the dedicated Square Root Calculator and Cube Root Calculator.