OnlineCalculator.xyz
Reviewed concrete example
A real nth root reverses an integer power
A root calculator looks for a real number y such that y raised to the integer index n equals the radicand x. The square root uses index two, the cube root uses three, and this page supports integer indices from two through one hundred.
The result is usually an approximation because many roots are irrational. The tool displays the result raised back to the selected index and an absolute residual. That check helps reveal rounding and numeric limits without claiming that every displayed digit is exact.
Even and odd indices have different negative domains
An even power of a real number cannot be negative. Therefore an even root of a negative real radicand has no real result. Enter −16 with index 2 and the page returns a real-domain message rather than NaN or an invented positive value.
An odd power preserves the sign. The cube root of −125 is −5 because (−5)^3 = −125. For any supported odd index, the calculator finds the root of the absolute value and restores the negative sign.
Complex numbers can express even roots of negatives, but complex mode is outside this page. The error links the limitation to the declared “real nth root” result.
Worked perfect roots
Enter 81 with index 4. The result is 3 because 3^4 = 81. The calculator tests guarded nearby integers and can label this as a verified perfect integer root. It still displays the substitution and verification.
Enter 64 with index 3. The cube root is 4. Enter 1 with any supported index and the root is 1. Enter zero and the result is zero. These simple cases are useful behavior checks.
For −32 with index 5, the real root is −2 because (−2)^5 = −32. Change the index to 4 and the real result becomes unavailable because the index is even.
Worked irrational root
Enter 2 with index 2. The square root is irrational. With six display places, the result is approximately 1.414214. Raising the unrounded internal result to the second power should return a value very close to 2; raising the displayed rounded value produces a small residual.
The result table distinguishes the internal verification from the copied rounded number. Increasing decimal places can reduce the rounding component of the residual, but browser floating-point precision sets a practical ceiling.
Another example is the cube root of 20, approximately 2.714418. The page does not try to express it as a simplified radical or symbolic form. It is a numeric real-root tool.
Root index is not a denominator typed anywhere
The nth root is sometimes written as x^(1/n), but computing that text with a generic power function can mishandle negative x in real arithmetic. The calculator branches by sign and parity instead of expecting a fractional exponent to return the desired odd negative root.
The index must be an integer. An index of zero has no ordinary nth-root meaning in this interface. An index of one would simply return the input and is excluded because it is not a root operation that needs a calculator.
Verification residual
The residual is |y^n − x| using the unrounded computed root y. A residual near zero shows numeric consistency with the equation. It is not an error estimate for a measurement and does not prove that the algorithm delivered every exact digit.
For large indices or magnitudes, raising the result back can amplify rounding or overflow. The page detects nonfinite verification and reports the browser limit. A perfect-power badge is restricted to safe guarded integer checks so a nearly integral floating result is not mislabeled.
Roots and units
Taking a root of a dimensional quantity changes units. The square root of an area in square meters can have meters as a unit under appropriate assumptions. The cube root of a volume can produce a length. This calculator accepts no unit and cannot verify dimensional reasoning.
Geometry, statistics, engineering, finance, and science may use roots inside larger formulas with constraints and uncertainty. A standalone numeric answer is not professional validation. Use the correct formula, units, and precision for the field.
Limits of this method
The page does not simplify radicals, factor integers, return complex roots, list all complex solutions, accept matrices, or compute symbolic expressions. It accepts one finite real radicand and one guarded positive integer index.
Extreme magnitudes can overflow or underflow. Irrational outputs are rounded for display. No arbitrary precision is promised, and values are not sent to an external mathematics engine.
Frequently asked questions
Why does a negative cube root work but a negative square root not work here?
Odd powers can be negative, while even powers of real numbers are nonnegative. The page operates only in the real-number domain.
Is a displayed root exact?
Only a safely verified perfect integer root is labeled exact. Other results are numeric approximations with a verification residual.
What does the residual mean?
It is the absolute difference between the radicand and the computed root raised back to the index under browser arithmetic.
Does the tool save radicands?
No. Inputs and results remain in current page state and are not uploaded or persisted.
Enter a radicand and integer index, calculate, and compare the approximate root with its raised-back verification before copying it.