Nth Root Calculator
Square root, cube root, fourth root, fifth root — or any custom n. Step-by-step work, exact perfect-power detection, honest handling of negatives.
Calculate ⁿ√x
Worked examples
| Input | Meaning | Result |
|---|---|---|
| √144 | square root of 144 | 12 (exact: 12² = 144) |
| √2 | square root of 2 | ≈ 1.41421356237 |
| ∛27 | cube root of 27 | 3 (exact: 3³ = 27) |
| ∛2 | cube root of 2 | ≈ 1.25992104989 |
| ∛(−8) | cube root of −8 | −2 (exact: (−2)³ = −8) |
| ∜81 | 4th root of 81 | 3 (exact: 3⁴ = 81) |
| ∜16 | 4th root of 16 | 2 (exact: 2⁴ = 16) |
| ⁵√32 | 5th root of 32 | 2 (exact: 2⁵ = 32) |
| ⁵√(−243) | 5th root of −243 | −3 (exact: (−3)⁵ = −243) |
| ⁶√64 | 6th root of 64 | 2 (exact: 2⁶ = 64) |
Frequently asked questions
What is an nth root?
The nth root of x, written ⁿ√x, is the number that gives x when raised to the power n. It is the same as the fractional exponent x^(1/n). Example: the 4th root of 81 is 3, because 3⁴ = 81.
How do I find the cube root of a number?
Take the number to the power 1/3. If it is a perfect cube the answer is exact: ∛27 = 3 because 3³ = 27. Otherwise you get a decimal: ∛2 ≈ 1.25992104989. The calculator above shows every step.
What is the fourth root of 81?
The fourth root of 81 is 3, because 3⁴ = 3 × 3 × 3 × 3 = 81. Similarly, the fourth root of 16 is 2, since 2⁴ = 16.
What is the fifth root of 32?
The fifth root of 32 is 2, because 2⁵ = 32. For an odd root you can also use negative numbers: the fifth root of −243 is −3, since (−3)⁵ = −243.
Can I take the root of a negative number?
Only with an odd root. ∛(−8) = −2 is perfectly valid because (−2)³ = −8. But an even root of a negative number has no real answer — √(−4) does not exist among real numbers, and this calculator tells you so instead of guessing.
How does the step-by-step work shown here?
The calculator rewrites ⁿ√x as the fractional exponent x^(1/n) — for a cube root, 1/3 = 0.333333333333 — then checks whether x is a perfect power using exact integer arithmetic, and finally evaluates the power with one Newton refinement step for full double precision.
What is a perfect power?
A perfect nth power is an integer that equals some integer k raised to n, for example 64 = 2⁶ and 10000 = 10⁴. When your input is one, the calculator returns the exact integer instead of a rounded decimal.
What is the difference between an nth root and a fractional exponent?
They are the same operation in the special case m = 1: ⁿ√x = x^(1/n). Fractional exponents in general allow any numerator, like x^(2/3). This tool focuses on roots; for general a^(m/n) use a dedicated exponent calculator.
Free tool by NRTechStudio