Quick answer
The n-th root of x is x^(1/n). The cube root of 27 is 3. Even roots of negative numbers are not real.
Enter a number and a root index. The calculator returns the root.
Related tools: Square Root Calculator: Square root with simplified radical form. Cube Root Calculator: Cube root of a number. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Root Calculator
- Enter number.
- Enter root (n) (a whole number, between 2 and 1000). It starts at 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Number | A number | — |
| Root (n) | A whole number | ≥ 2, ≤ 1000, starts at 3 |
Formula and method
Formula
ⁿ√x = x^(1/n)
IEEE-754 double precision; shown to at most 6 decimal places.
Odd roots of negative numbers are real; even roots of negative numbers are rejected.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Number = 27; Root (n) = 3.
| Result | Value |
|---|---|
| 3th root of 27 | 3 |
Example 2
Inputs: Number = 16; Root (n) = 4.
| Result | Value |
|---|---|
| 4th root of 16 | 2 |
Example 3
Inputs: Number = -8; Root (n) = 3.
| Result | Value |
|---|---|
| 3th root of -8 | -2 |
The first line is the headline result (3th root of 27). IEEE-754 double precision; shown to at most 6 decimal places.
More worked cases
√16 = 4, ∛64 = 4, the fourth root of 81 is 3 and the fifth root of 32 is 2. Roots of non-perfect powers are irrational, such as ∛10 ≈ 2.1544.
Edge cases
Even roots of negative numbers (such as √−4) are not real numbers. Odd roots of negatives are real: ∛(−27) = −3. Roots of numbers between 0 and 1 are larger than the number.
Choosing the index n
The nth root of x is x^(1/n). The cube root of 27 is 3, the 4th root of 81 is 3 because 3⁴ = 81, and the 5th root of 32 is 2.
In real numbers, an odd root of a negative number exists (the cube root of −8 is −2), but an even root of a negative number does not, because no real number raised to an even power is negative.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Number: entering "-4" shows "An even root of a negative number is not a real number." beside that field and no result.
Choosing between the Root Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Square Root Calculator](/math/square-root-calculator/) | number | √144 |
| [Cube Root Calculator](/math/cube-root-calculator/) | number | Cube root |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Average Calculator](/math/average-calculator/) | numbers | Average (mean) |
Background: rounding and basic arithmetic
Rounding to a place looks at the digit just to its right: 0 to 4 rounds down and 5 to 9 rounds up (the half-up convention, so 2.5 becomes 3). Rounding 838.274 gives 800 to the nearest hundred, 840 to the nearest ten and 838 to the nearest one.
Some systems use banker's rounding (half to even), which sends 2.5 to 2 and 3.5 to 4; check which convention your task requires, and round only at the end of a multi-step calculation.
Assumptions and limitations
- Even root of a negative number is not real
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for number to confirm this.
Frequently asked questions
What is the 4th root of 16?
The 4th root of 16 is 2, since 2⁴ = 16.
Is a root the same as a power?
It's a fractional power.
How do I enter a square root?
Enter 2 as the root index; for the cube root enter 3.
What is the 5th root of 32?
It is 2, because 2⁵ = 32.
Is the nth root the same as raising to the power 1/n?
Yes. The nth root of x equals x^(1/n), so the cube root is the 1/3 power.
What does the Root Calculator calculate?
Nth root of a number. It returns 3th root of 27.
What formula does the Root Calculator use?
ⁿ√x = x^(1/n). Odd roots of negative numbers are real; even roots of negative numbers are rejected.
What do I need to enter in the root Calculator?
You enter Number and Root (n).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Even root of a negative number is not real.
What is an example of a root result?
With number 27 and root (n) 3, the 3th root of 27 is 3.
Related calculators
- Square Root Calculator — Square root with simplified radical form.
- Cube Root Calculator — Cube root of a number.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Average Calculator — Arithmetic mean, sum, count, minimum and maximum of a list of numbers.
- Long Division Calculator — Divide whole numbers to a chosen number of decimal places.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.