Nth Root Calculator
Calculate any k-th root of n instantly, shown as n^(1/k). Odd roots of negatives work; even roots of negatives are explained as undefined for real numbers.
Nth Root
4√81
3
Exact integer result
Number and Root Index
The number to take the root of.
2 = square root, 3 = cube root, 4, 5, …
How the Calculation Works
The k-th root is computed as the fractional power n^(1/k). For negative n with odd k, the calculator factors out the sign and applies the power to |n|, returning the negative root — this avoids the NaN that naive fractional powers produce for negatives. Perfect k-th powers are detected and returned as exact integers (4√625 = 5); irrational results are rounded to 9 significant digits. Newton's method on x^k − n = 0, iterating x → ((k−1)x + n/x^(k−1)) / k, converges to the same value.
Try These
Quick Answer
The k-th root of n is the number x where x^k = n, written ⁿ√x or n^(1/k). Enter n and the root index k to get the result instantly: 4th root of 81 = 3, 5th root of 32 = 2, cube root of -27 = -3. Odd-index roots accept negative inputs; even-index roots of negative numbers are undefined for real numbers. Non-perfect powers return the root to 9 significant digits.
Key Facts
- ⁿ√x = x^(1/k): the k-th root is the 1/k power (ⁿ√x)^k = x
- Even k (2nd, 4th, 6th, …): negative inputs are undefined for real numbers
- Odd k (3rd, 5th, 7th, …): negatives work — 5th root of -32 = -2
- Each positive number has two real even roots (±ⁿ√x); the radical means the positive one
- Each number has exactly one real odd root, with the same sign as the input
- ⁿ√(x·y) = ⁿ√x · ⁿ√y and x^(m/k) = (ⁿ√x)^m for integer m
- Root index must be ≥ 2; k = 1 returns the number itself
- ⁿ√x shrinks toward 1 as k grows: 10th root of 1024 ≈ 2.0000
Frequently Asked Questions
The k-th root of n (also written with index k under the radical, k√n) is the number x satisfying x^k = n. For example, the 4th root of 81 is 3 because 3⁴ = 81, and the 5th root of -32 is -2 because (-2)⁵ = -32. It is exactly the fractional power n^(1/k), so nth-root questions are exponent questions in disguise.
Raising to an even power never yields a negative: (±x)² , (±x)⁴ are always ≥ 0, because the sign cancels in pairs. So no real number satisfies x² = -16 or x⁴ = -81. Odd powers keep the sign — (-2)³ = -8 — which is why odd roots of negatives exist. Over the complex numbers even roots do exist (using i = √-1), with k distinct complex values.
Use Newton’s method on x^k − n = 0: iterate x → ((k−1)·x + n/x^(k−1)) / k. Start from a rough guess; each iteration roughly doubles the correct digits. Example, 5th root of 32 starting at 2 is already exact. Alternatively use logarithms: n^(1/k) = 10^(log₁₀(n)/k) — divide the log by k, then undo the log.
Even roots (square, 4th, 6th, …) of positive numbers have two real values (±), and the radical symbol conventionally returns the principal (positive) one; they are undefined for negative reals. Odd roots (cube, 5th, 7th, …) have exactly one real value for every input, with the same sign as the input — the 3rd root of -27 is -3, full stop.
Identically: the k-th root of n equals n^(1/k), and more generally x^(m/k) = (k-th root of x)^m. This follows from the power rule (x^a)^b = x^(a·b): raising n^(1/k) to the k-th power gives n. That equivalence is why this calculator shows every result both in radical notation (ⁿ√x) and exponent notation (x^(1/k)).
Inputs that are perfect k-th powers return exact integers (4th root of 625 = 5, detected when the computed root is an integer). All other results are computed at full double precision and displayed to 9 significant digits — e.g., 3rd root of 10 = 2.15443469. Like all irrational roots, these are approximations: the true decimal never terminates.
Nth Root
4√81
3
Exact integer result