Find the nth root of any number with domain validation for even roots of negative values and exact-result detection.
Last updated
Quick presets
Result
Square root of 9 (exact).
How it works
The nth root of a number x is the value r such that rn = x. Computed as x1/n.
Algebra
The root calculator finds the nth root of any number. Enter a value and a root degree to get the result instantly, with detection of exact integer results and domain warnings for even roots of negative numbers.
The nth root of a number x is the value y such that y raised to the power n equals x. For example, the 4th root of 81 is 3 because 3^4 = 81. Square roots (n = 2) and cube roots (n = 3) are the most common cases.
Even roots of negative numbers have no real solution — the square root of -4, for instance, is not a real number. Odd roots of negative numbers are valid: the cube root of -8 is -2 because (-2)^3 = -8.
y = x^(1/n)
The nth root of x equals x raised to the power 1/n.
This calculator handles real-valued roots only. Complex roots of negative numbers under even degrees are flagged but not computed. For very large values, floating-point precision may affect the last digits.
A worked example helps translate the nth root calculator with domain validation maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.
That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.
Frequently asked questions
A root is the inverse of a power. If 2^3 = 8, then the cube root of 8 is 2. Taking the nth root is equivalent to raising to the power 1/n.
Not in real numbers. The square root of -1 is the imaginary unit i. This calculator works with real numbers only and will flag even roots of negatives as out of domain.
The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.
Also in Functions
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Find the real cube root of any number with perfect-cube checks, nearby cube references, exact radical simplification, negative-number support.
Calculate the square root of any non-negative number with principal and negative roots, equation checks, and nearby perfect-square references.
Calculate any power or exponent — raise a base to any power, find roots, and switch between power and root modes.
Evaluate a real number, fraction, decimal, or expression, then see the simplified value before absolute value, the non-negative result, number-line distance.
