Example 1
- x = 27
- n = 3
3
What is ∛27? 3.
Nth root calculator
Enter the radicand x and a non-zero integer degree n. The result is the principal real root y such that yⁿ = x, i.e. ⁿ√x = x^(1/n) for x > 0. With 27 and n = 3 you get 3; with 16 and n = 2 — 4; with 81 and n = 4 — 3. With odd n, negative x is allowed: ∛(−27) = −3. Even n with negative x, and n = 0, return an error. The calculator does not return complex roots.
Result
Enter x and a non-zero integer n.
n must be a non-zero integer. Even n requires x ≥ 0. Odd n allows negative x. n = 0 is undefined. The result is the principal real root, not the complex principal power.
How it works
x27
n3
ⁿ√x3
The nth root of a number x is a number y such that yⁿ = x. The notation ⁿ√x means that y. For x ≥ 0 there is exactly one non-negative principal real root: ⁿ√x = x^(1/n). The square root is the case n = 2 (√16 = 4). The cube root is n = 3 (∛27 = 3). That choice is the school-algebra standard and is not confused with a negative square root: (−4)² = 16, but √16 = 4.
The domain depends on whether n is even or odd. For even n the radicand must satisfy x ≥ 0 — otherwise there is no real root and the calculator reports an error (for example √(−16)). For odd n, negative x is allowed: ∛(−27) = −3, because (−3)³ = −27. Degree n is a non-zero integer; n = 0 is undefined because the exponent 1/n does not exist.
The calculator uses the numbers you enter in the fields x (pierwiastek-n-tego-stopnia-a) and n (pierwiastek-n-tego-stopnia-b). For x > 0 the result is x^(1/n), which matches exp((1/n) ln x) from NIST DLMF §4.2. For x < 0 and odd n the result is −Math.pow(−x, 1/n) — the real odd root, not the complex principal power (which for (−8)^(1/3) is often complex). Comma and period in the fields are equivalent. Header currency and unit labels do not enter the formula.
The result panel shows ⁿ√x, echoes of x and n, the power form, the check yⁿ, and — when x > 0 — a comparison with exp((1/n) ln x). The bar diagram compares x with yⁿ on a shared scale. Worked examples: 27 and 3 give 3; 16 and 2 give 4; 81 and 4 give 3; −27 and 3 give −3; −16 and 2 give an error; √8 ≈ 2.828427.
The separate square-root page computes only n = 2. A power with exponent 1/n is the other writing of the same relation, and for x > 0 it yields the same number. This calculator does not simplify radicals to a form like 6√2 and does not return complex roots — it stays with the principal real root.
Formula
ⁿ√x = x1/n for x ≥ 0 and integer n ≠ 0 (when n < 0: ⁿ√x = 1 / |n|√x, x ≠ 0).
For x > 0: ⁿ√x = exp((1/n) ln x) (NIST DLMF §4.2).
For x < 0 and odd |n|: ⁿ√x = −(|x|)1/|n| (real odd root).
For x < 0 and even |n|: no real root.
Symbols: x — radicand (the “x (radicand)” field); n — degree (the “n (degree)” field); y — result ⁿ√x. Mathematical model: principal real nth root, per school algebra and NIST DLMF §4.2 (powers and logarithm).
Theory, domain, and the power link
Definition
The nth root of a number x, written ⁿ√x or x^{1/n}, means in the reals the principal real value y such that yⁿ = x, with a branch choice: for x > 0 one takes y > 0; for x < 0 only odd n are allowed, with y < 0. The degree n is a nonzero integer. The calculator computes |x|^(1/|n|) with sign, and for x > 0 also shows exp((1/n) ln x).
NIST DLMF §4.2 (powers) and analysis texts define x^{1/n} = exp((1/n) ln x) for x > 0. MathWorld (Nth Root) stresses that an even root of a negative x has no real value. This calculator reports an error in that case — matching the ℝ model.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Historical and conceptual context
Square and cube roots were studied from antiquity (Babylon, Greece); the generalization to the nth root connects to equation theory and field extensions. In the complex numbers every x ≠ 0 has n roots; the principal Arg branch selects one. Here we stay with the real root.
Conceptually n = 0 is excluded (x^{1/0} undefined). Negative n means the reciprocal of the |n|th root: x^{−1/|n|} = 1/x^{1/|n|} when x ≠ 0.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Formal properties
For x > 0, n ∈ ℤ \ {0}: ⁿ√x = exp((ln x)/n) when n > 0; for n < 0 the result is 1/ⁿ√x with absN = |n|. For x = 0 and n > 0: ⁿ√0 = 0; for n < 0: error (division by zero).
For x < 0: if |n| is odd, ⁿ√x = −(|x|)^{1/|n|}; if |n| is even — no y ∈ ℝ. Identity: (ⁿ√x)ⁿ = x on the domain where the result exists.
For x > 0: ⁿ√(xᵐ) = x^{m/n} under matching positive branches.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Full derivation
Step 1 (x > 0). Seek y > 0 with yⁿ = x. Taking ln: n ln y = ln x ⇒ ln y = (ln x)/n ⇒ y = exp((ln x)/n).
Step 2 (x < 0, n odd). Let y = −z, z > 0. Then yⁿ = −zⁿ = x ⇒ zⁿ = |x| ⇒ z = |x|^{1/n}, y = −z.
Step 3 (x < 0, n even). yⁿ ≥ 0 for y ∈ ℝ, so it cannot equal negative x — hence an error.
Step 4 (n < 0). x^{1/n} = 1/x^{1/|n|} when x ≠ 0, provided the |n|th root exists.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Special cases
Examples: ³√27 = 3; √16 = 4; ⁴√81 = 3; ³√(−27) = −3; √(−16) → error; ⁵√1 = 1; ⁵√0 = 0. ²√8 = 2√2 ≈ 2.828.
n = 1 yields y = x. n = −1 yields 1/x when x ≠ 0. Very large |n| drive ⁿ√x toward 1 for x > 0.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Numerical and computational notes
x^(1/n) for x > 0 can be less stable than exp(ln x / n); the calculator shows both forms when finite. Comma and period are equivalent. n must be a nonzero integer.
The calculator does not return complex roots or all n roots on the circle.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
Related concepts and authorities
Related: exp, ln, log_a, square of a sum (a different model). For a square root, n = 2 suffices.
Authorities: NIST DLMF §4.2; Abramowitz & Stegun, Chapter 4; MathWorld — Nth Root; Wikipedia — Nth root. Educational tool.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Comparisons with classical tables (Abramowitz and Stegun) and with the NIST DLMF are meaningful for moderate values; near domain edges ulp differences are expected and do not indicate a conceptual defect of the calculator. MathWorld and Concrete Mathematics (Knuth) remain reference points for the definitions and notation used in the sections above.
How to use
- Enter x in the radicand field, for example 27.
- Enter a non-zero integer n, for example 3.
- Click Calculate or wait for the live result. 27 and 3 give 3. 16 and 2 give 4. 81 and 4 give 3.
- Check the panel: power form, yⁿ check, exp/ln when x > 0, and the bar diagram.
- −27 and 3 give −3. −16 and 2 give an error. n = 0 does not pass. A square root alone also lives on the square-root page.
Symbol glossary
ⁿ√x is a y such that yⁿ = x. ∛27 = 3. √16 = 4. With odd n, negative x yields negative y.
- ⁿ√x
- Principal real root. Non-negative when x ≥ 0. ∛27 = 3. √16 = 4.
- x
- Radicand in the “x (radicand)” field. For even n: x ≥ 0.
- n
- Degree in the “n (degree)” field. Non-zero integer. 2 = square, 3 = cube.
- x^(1/n)
- Power form for x > 0; equivalent to exp((1/n) ln x).
Examples
Example 2
- x = 16
- n = 2
4
What is √16? 4.
Example 3
- x = 81
- n = 4
3
What is ⁴√81? 3.
Example 4
- x = −27
- n = 3
−3
What is ∛(−27)? −3.
Example 5
- x = −16
- n = 2
error
Does √(−16) have a real result? No — error.
Example 6
- x = 8
- n = 2
≈ 2.828427
What is √8? ≈ 2.828427.
Example 7
- x = 1
- n = 5
1
What is ⁵√1? 1.
Example 8
- x = 0
- n = 5
0
What is ⁵√0? 0.
Related calculators
Common questions
What is the domain of the nth root?
Degree n is a non-zero integer. For x ≥ 0 a real result exists for every such n. For x < 0 a real result exists only when |n| is odd. Even n with negative x returns an error — there is no real root. n = 0 is undefined.
What is ∛27?
The result is 3, because 3³ = 27. That is the first example on this page and the same result after “Fill this example”.
What are √16 and ⁴√81?
√16 = 4 (n = 2) and ⁴√81 = 3 (n = 4). The second field is the degree n, so the same calculator computes square, fourth, and other integer roots.
Does a negative x work?
With odd n, yes: ∛(−27) = −3. With even n, no — √(−16) does not exist in the reals, and the calculator reports an error instead of a blank or zero.
What if n = 0?
A root of degree 0 is undefined because the exponent 1/n does not exist. The calculator shows an error; it no longer returns a misleading zero.
How is this different from the square-root calculator?
On the square-root page n is always 2 and there is one field. Here you enter x and any non-zero integer n. With n = 2 and x = 16 the result is still 4.
How is the root written as a power and with exp/ln?
For x > 0, ⁿ√x = x^(1/n) = exp((1/n) ln x). The exp/ln form follows the real power definition in NIST DLMF §4.2. For negative x with odd n the calculator uses −(|x|)^(1/n), not the complex principal power.
What is √8?
About 2.828427. √8 is not an integer. The panel also shows the check y² ≈ 8.
Can the result be negative?
Yes when x < 0 and n is odd: ∛(−27) = −3. With n = 2 the result is non-negative: √16 = 4, not −4, even though (−4)² = 16.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on x and n. The currency and unit settings in the page header do not affect this result.
Does a comma in 27.0 work?
Yes. 27 and 27.0 at n = 3 give 3. Comma and period are equivalent.
What about x = 0?
For n > 0 the result is 0. For negative n a root of zero is undefined (division by zero).
How do I calculate an nth root?
Enter x and a non-zero integer n. For x > 0 the result is x^(1/n), equivalently exp((1/n) ln x). With 27 and n = 3: 27^(1/3) = 3, because 3³ = 27. The panel also shows the check yⁿ.
What is ⁵√32?
2, because 2⁵ = 32. Enter x = 32 and n = 5. That is a classic fifth-root example.
Does the calculator return complex roots?
No. The result is only the principal real root. With even n and negative x you get an error instead of i or other values in ℂ.
What is ⁴√16?
2, because 2⁴ = 16. Enter x = 16 and n = 4. The principal root is non-negative — not −2.
How does ⁿ√x differ from the power x^n?
A root is the inverse of a power: y = ⁿ√x satisfies yⁿ = x. The power x^n multiplies x by itself n times. With x = 27 and n = 3 the root is 3, while 27³ = 19,683.
Knowledge sources
The definition, domain, and exp/ln form below come from recognized mathematical sources.
- NIST DLMF §4.2 — Logarithm, exponential, powers
- MathWorld — Principal Root
- Wikipedia — Nth root (domain, even/odd n)
Page Updated November 2, 2026.