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Root Calculator

Calculate a real principal nth root from degree 2 through 100 and verify it by exponentiation.

Privacy: your files never leave your device. All processing happens locally in your browser.

How to use

  1. 1.Enter a finite radicand, including a negative value or scientific notation if needed.
  2. 2.Choose an integer root degree from 2 through 100 and review the real-domain message or result.
  3. 3.Check the displayed inverse power and relative error, then copy the rounded result if its precision is sufficient.

About Root Calculator

Root Calculator finds the real principal nth root of a finite number for any whole-number degree from two through one hundred. Enter a radicand using ordinary decimal or scientific notation and choose the root degree. The result updates immediately, displays up to twelve significant digits, and includes an inverse check that raises the result back to the selected degree. Calculation and copying remain entirely in the browser, with no account, network request, or stored history.

The definition is direct: if b raised to the integer n equals a, then b is an nth root of a. This tool reports the principal real root. For a nonnegative radicand, the principal result is nonnegative. A negative radicand has a real result only when the degree is odd. For example, the third root of minus eight is minus two because minus two cubed is minus eight. The fourth root of minus sixteen has no real value, so the page returns a clear domain error instead of NaN or a misleading positive result.

Degree must be an integer from two through one hundred. A degree of one would simply return the input and is outside this tool's useful scope. Zero, negative, fractional, empty, and excessively large degrees are rejected. The radicand must be a finite JavaScript Number with absolute magnitude no greater than 1e300. That conservative limit leaves room for exponentiation checks inside the approximate floating-point range while still covering ordinary scientific, engineering, and educational examples.

For a nonzero radicand, the calculator takes the absolute value to the reciprocal of the degree. If the original radicand is negative and the degree is odd, it restores the negative sign after calculating the positive magnitude. This matters because directly asking JavaScript for a negative base raised to a fractional exponent returns NaN, even when the mathematical odd root is real. Separating magnitude and sign avoids that runtime trap while preserving the real-domain rule.

The inverse check raises the computed root to the integer degree and compares it with the original radicand. A relative error is displayed in scientific notation. Perfect-power examples often round back exactly or extremely closely, while irrational roots show small binary floating-point differences. The check is evidence about numerical consistency, not a symbolic proof. Inputs and results are IEEE 754 double-precision approximations, so very large, tiny, or ill-conditioned cases should be verified with arbitrary-precision software when exact digits matter.

Display formatting uses twelve significant digits and removes trailing zeros. This makes the square root of two readable as 1.41421356237 instead of exposing a long implementation tail. Scientific notation may appear for very large or small values. Formatting changes the visible string, not the internal value used for the inverse check. Copy result writes exactly the displayed root. If clipboard access is denied, the value remains visible and selectable and the status does not falsely claim success.

This broader nth-root tool differs from the dedicated Cube Root Calculator. The existing cube-root page offers a focused one-input workflow and the browser's specialized Math.cbrt behavior. This page handles arbitrary degrees, including fourth, fifth, sixth, and high odd roots, and must therefore enforce the even-versus-odd real-domain distinction. If you only need repeated cube-root work, the focused related tool may be faster to use.

The calculator returns one real principal value, not every solution to an equation. For example, four is the principal square root of sixteen, even though both four and minus four solve x squared equals sixteen. It does not calculate complex roots, list all roots of unity, simplify radicals symbolically, factor perfect powers, rationalize denominators, solve equations, or preserve exact fractions. Use a symbolic algebra system when those outputs are required.

Do not use rounded output as the sole basis for safety-critical engineering, financial settlement, cryptography, or scientific publication. Check units before and after the calculation: taking a root changes dimensional exponents, and a bare number cannot detect a mismatched unit model. The page has no hidden unit conversion and does not infer whether the radicand represents area, volume, probability, power, or an abstract exercise.

OpenStax defines principal nth roots, supplies worked fourth- and fifth-root examples, and explains that even roots of negative numbers are not real while odd roots are real. A second OpenStax algebra text independently cross-checks that rule. ECMA-262 documents Number exponentiation as an implementation-approximated operation, which supports the disclosed floating-point limitation. Eight external golden cases cover positive, negative odd, fractional, scientific-notation, and high-degree inputs, while independent boundary tests reject invalid real-domain and degree combinations.

Methodology & sources

Validate a finite bounded radicand and integer degree, reject negative radicands with even degree, calculate the positive magnitude using reciprocal exponentiation, restore sign for negative odd roots, format to 12 significant digits, and expose an inverse relative-error check.

Frequently asked questions

Why is an even root of a negative number rejected?
No real number raised to an even integer power is negative. Odd degrees can preserve a negative sign and therefore have a real result.
Does a square root have two answers?
The radical symbol denotes the nonnegative principal root. An equation such as x squared equals 16 has two solutions, but this calculator reports the principal root 4.
Why does the inverse check show a tiny error?
JavaScript Number arithmetic is binary floating point. Irrational roots and many decimal values cannot be represented exactly, so exponentiation can differ by a small rounded amount.
Can it calculate complex or symbolic roots?
No. It returns one approximate real principal root and does not list complex roots, simplify radicals, or preserve exact algebraic forms.

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