nthroots

An nth root of a number a is any number b such that b^n = a. It is usually written as √[n]{a} or a^(1/n). The radical notation is most common for real numbers, while the exponent form extends naturally to complex numbers. When a is nonnegative, the principal nth root is the unique nonnegative real root.

In the real numbers, the situation depends on n. If n is odd, every real a has exactly one real nth root. If n is even, a < 0 has no real nth root, a = 0 gives b = 0, and a > 0 yields two real roots, b = ±√[n]{a}. The principal nth root of a > 0 is the positive one.

In the complex plane, every nonzero a has exactly n distinct nth roots. If a = r e^{iθ} with r > 0 and θ chosen modulo 2π, the nth roots are r^{1/n} e^{i(θ + 2πk)/n} for k = 0, 1, ..., n−1.

Key properties include (a^m)^(1/n) = a^(m/n) when defined, and, for nonnegative a and b, √[n]{ab} = √[n]{a} √[n]{b} in the real setting. In the complex setting, branch choices and multiple values require careful handling.

Common examples are the square root √a and the cube root ∛a. Complex roots of unity (such as the four fourth roots of 1) illustrate the multiplicity of nth roots in the complex plane.