Root Calculator
Square root, cube root, or any nth root — handles negative radicands with odd index correctly.
Inputs
Can be negative if the index is odd.
Must be a positive integer ≥ 2.
Root
2.0000000000
Index used
5
Step by step
Compute 5√32
32^(1/5)
= 2
Verify: raise the result back to the index
(2)^5
= ≈ 32 ✓
How it works
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
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.