Root Calculator
Find square roots, cube roots and any nth root of a number.
About Root Calculator
The nth root of a number x is the value r such that rⁿ = x. The square root (n = 2) and cube root (n = 3) are the most common. For a negative radicand, whether a real answer exists depends on the index: an odd index (3, 5, 7 …) produces a real negative root (e.g. ∛(−8) = −2); an even index produces a complex (imaginary) number (e.g. √(−4) = 2i).
Formulas
Nth root
nth root of x = x^(1/n)
- x
- Radicand
- n
- Index (positive integer ≥ 2)
Negative radicand (odd index)
n-th root of (−x) = −(n-th root of x), when n is odd
Worked example
With the default values below, Root Calculator returns Root of 2.0000000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Root typeone of Square root (√), Cube root (∛), Nth root
- Nth root
- Radicand (number under the root)
- 32
- Index (n)minimum 2
- 5
Results
- Root
- 2.0000000000
- Index used
- 5
Step by step
- 1Compute 5√32= 2
- 2Verify: raise the result back to the index= ≈ 32 ✓
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Why is the square root of a negative number imaginary?
Any real number squared is non-negative, so no real number r satisfies r² = −4. The solution exists in the complex numbers: √(−4) = 2i, where i = √(−1). The cube root of a negative number IS real because (−2)³ = −8.
What is the 5th root of 32?
2, because 2⁵ = 32.
Is √9 exactly 3 or approximately 3?
Exactly 3. Perfect squares have exact integer roots. Non-perfect squares like √2 are irrational and cannot be expressed as a fraction — they continue infinitely without repeating.