Find square roots, cube roots and any nth root of a number.
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).
Nth root
nth root of x = x^(1/n)
Negative radicand (odd index)
n-th root of (−x) = −(n-th root of x), when n is odd
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.
2, because 2⁵ = 32.
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.