An nth root is the number that, when raised to the integer degree n, reproduces the original radicand — for example, the fifth root of 32 is 2 because 2 to the power of 5 equals 32. To calculate a root, you supply two values: the radicand (the number under the radical) and the integer degree n (the index of the root), and the calculator returns the principal real nth root. For nonnegative radicands, the principal root is the nonnegative value; for negative radicands, a real result exists only when n is odd. The Root Calculator handles integer degrees from 2 through 100, shows up to twelve significant digits, and includes an inverse check that raises the result back to the chosen degree so you can verify the answer without leaving the browser.
The general definition is direct: if b raised to the integer n equals a, then b is an nth root of a. A radicand can have more than one real root when the equation b to the n equals a is solvable in more than one way — most commonly for even degrees where both a positive and a negative value satisfy the equation. The radical symbol, however, always refers to the principal root, which is the nonnegative value for nonnegative radicands and the unique real value for negative radicands at odd degrees. This convention is consistent across standard algebra references, including the OpenStax Elementary Algebra treatment of higher roots and the OpenStax College Algebra chapter on radicals and rational exponents.
How Root Calculation Works Mathematically
Behind every root computation is the relationship b to the n equals a. Once you choose a radicand a and a positive integer degree n, the calculator searches for the value b that, when multiplied by itself n times, equals a. The simplest closed form for the principal root takes the absolute value of a to the reciprocal of n, then restores the original sign when a is negative and n is odd. For example, the seventh root of 128 is 2 because 2 to the power of 7 equals 128, while the seventh root of negative 128 is negative 2 because negative 2 to the power of 7 equals negative 128. Irrational radicands such as 2 or 7 produce non-terminating decimal expansions like 1.41421356237 for the square root of 2; these cannot be written exactly in any finite decimal form, so the calculator displays a rounded approximation and flags the precision through the inverse check.
Root computation also appears in unit analysis. Taking a root changes the dimensional exponent on a quantity, so the square root of an area in square meters yields a length in meters, and the cube root of a volume in cubic centimeters yields a length in centimeters. A bare number never tells the calculator what units the radicand represents, which is why you should always confirm the input and output carry consistent units before using a root value in a larger equation.
How to Calculate a Root Step by Step
- Enter a finite radicand in the input field, using ordinary decimal form for typical values or scientific notation (such as 1.5e12 or negative 3.2e negative 5) for very large or very small numbers, including negative values when a real result might exist.
- Choose an integer root degree from 2 through 100 in the degree selector and review the page's real-domain message or the displayed result — the calculation runs immediately as inputs change.
- Read the displayed inverse power and relative error to confirm the answer, then copy the rounded result using the Copy button if the displayed precision is sufficient for your task.
Each input is validated as you type. A radicand outside the finite JavaScript Number range or above an absolute magnitude of 1e300 is rejected because the inverse exponentiation check would overflow the floating-point representation; a degree outside the integer range 2 through 100 is rejected as well. If you submit a negative radicand with an even degree, the page returns a clear domain error instead of producing a misleading positive result or the NaN value JavaScript would return for fractional exponents on negative bases.
Real-Domain Rules: When a Root Exists
The presence or absence of a real nth root depends on the sign of the radicand and whether the degree is even or odd. The table below summarizes the real-domain behavior enforced by the Root Calculator and consistent with the OpenStax algebra references.
| Radicand sign | Degree parity | Real root exists? | Result direction |
|---|---|---|---|
| Nonnegative | Any integer 2 to 100 | Yes | Nonnegative principal value |
| Negative | Even | No | Domain error (no real value) |
| Negative | Odd | Yes | Negative principal value |
| Zero | Any integer 2 to 100 | Yes | Exactly zero |
The third row is the most useful in practice. Because an odd integer power preserves sign, raising a negative number to an odd degree reproduces a negative radicand, so a real negative root always exists for odd n. An even integer power, on the other hand, always produces a nonnegative result, which means even-degree roots of negative numbers have no real value at all. Asking JavaScript directly for a negative base raised to a fractional exponent returns NaN, so the calculator avoids that runtime trap by computing the positive magnitude first and restoring the negative sign only when the degree is odd.
Reading the Inverse Check and Relative Error
Every result on the page is paired with an inverse check: the displayed root raised to the chosen degree and compared with the original radicand. The page shows the inverse power as a separate value and a relative error in scientific notation. Perfect-power radicands — values whose principal root is a small integer, like the fourth root of 81, which equals 3 — typically round back exactly or to a relative error at the limit of double-precision. Irrational roots such as the seventh root of 9 show a small but nonzero relative error because binary floating-point cannot represent the exact decimal expansion of those values. The check is evidence about numerical consistency rather than a symbolic proof; if you need exact algebraic forms, you should switch to a symbolic algebra system.
The relative error is the magnitude of the difference between the radicand and the inverse power, divided by the radicand. A value displayed as 1.4e negative 15 means the inverse check agrees with the input to roughly fifteen significant digits, which is the practical limit of IEEE 754 double-precision arithmetic. Inputs and results are stored as double-precision approximations throughout, so very large, very small, or numerically ill-conditioned cases should be cross-checked with arbitrary-precision software whenever exact digits matter. This disclosed limitation is consistent with the ECMA-262 description of Number exponentiation as an implementation-approximated operation.
Input Limits and What the Tool Rejects
The radicand must be a finite JavaScript Number with absolute magnitude no greater than 1e300. That conservative limit leaves room for the inverse exponentiation check inside the approximate floating-point range while still covering ordinary scientific, engineering, and educational examples. Degrees must be whole numbers between 2 and 100 inclusive; the value 1 is excluded because the first root would simply return the input, and zero, negative, fractional, empty, and excessively large degrees are all rejected with a clear error message. Display formatting uses twelve significant digits and removes trailing zeros, so the square root of 2 reads as 1.41421356237 instead of exposing a long implementation tail, and scientific notation appears for very large or very small values without changing the underlying double-precision value used by the inverse check. All calculation and copying happen locally in your browser — there is no account, no network request for the math, and no stored history of inputs or results.
Worked Example: Third Root of Negative 8
Suppose you want the third root of negative 8. You enter the radicand negative 8, select degree 3, and the page returns negative 2. The inverse check raises negative 2 to the third power and reports negative 2 to the power of 3 equals negative 8, with a relative error of 0 because the result is exactly representable in binary floating point. By contrast, entering the radicand negative 16 with degree 4 produces a domain error rather than a misleading positive value, because no real number raised to an even integer power equals a negative number. This single contrast captures the even-versus-odd real-domain rule and shows why the calculator refuses to fake an answer.
When a Different Calculator Fits Better
The Root Calculator returns one approximate real principal root. It does not list every solution to an equation, calculate complex roots, simplify radicals symbolically, factor perfect powers, rationalize denominators, or preserve exact fractions. Use a computer algebra system when those outputs are required. For repeated cube-root work, the focused Cube Root Calculator page offers a single-input workflow built around the browser's specialized cube-root behavior and may be quicker to use. For verifying that a candidate root is correct through symbolic manipulation, a CAS or hand derivation is the right tool. A rounded numeric output should also not be the sole basis for safety-critical engineering, financial settlement, cryptography, or scientific publication — confirm units before and after the calculation and reproduce the computation with arbitrary-precision software when the digits must be exact.