Number and Root Index
The number to take the root of.
2 = square root, 3 = cube root, 4, 5, …
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Your Results
Results are also pinned to the bottom of your screen.Nth Root
4√81
3
Exact integer result
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.
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
What does the nth root mean?
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.
Why can’t even roots of negative numbers be real?
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.
How do I calculate a root by hand?
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.
What is the difference between an even root and an odd root?
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.
How is the result related to fractional exponents?
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)).
What precision do results have?
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.