Nth Root Calculator

Formula

Nth root = Number^(1 ÷ Root n)

Worked example

With Number: 81, Root n: 4, nth root is 3.

Calculation method

The rule is Nth root = Number^(1 ÷ Root n).

For this nth root example, changing number from 81 to 82 while keeping the other inputs fixed changes nth root from 3 to 3.009217.

Reading nth root

The principal square root is nonnegative. A cube root can be negative when its input is negative. In the worked values on this page, number 81, root n 4 produce nth root 3.

Practical use

Use roots to recover a dimension from a squared or cubed quantity or solve a power relationship. The specific task here is to find nth root from number, root n.

Limits and common mistakes

An even root of a negative number is not a real number. For a general nth-root calculation, keep the requested degree and domain clear. Verify number before interpreting nth root; a value measured on another basis would answer a different question.

Why the relationship works

This page raises a nonnegative input to the reciprocal of the entered positive root degree. If the degree is two, the result is the principal square root; if it is three, the result is the nonnegative cube root. It returns one principal real value, not every solution of a related equation. The nth root rule uses number, root n in Nth root = Number^(1 ÷ Root n); read each symbol as the field named on this page.

Interpreting another set of values

For nth root, the fields labeled number, root n are the known quantities. The calculator solves for nth root using Nth root = Number^(1 ÷ Root n). The result may be an exact integer, a rounded decimal, an exponent, or a principal root. Preserve the original fraction or expression when a rounded display would lose precision needed later. When comparing two calculations, keep their units and input definitions identical.

For a second nth root check, keep the other default inputs fixed and change number from 81 to 83. Nth root changes from 3 to 3.018349. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

This calculator accepts a nonnegative number and a positive root degree. A zero or negative degree is outside its stated input range. Rounding a root before raising it back to the original power can create a small difference. The reported nth root is 3 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.