evenroot

Evenroot is a mathematical term used to describe the nth root of a number where the index n is even (for example, n = 2, 4, 6). The most common example is the square root, denoted by sqrt(x) or x^(1/2). A fourth root is written as sqrt[4](x) or x^(1/4).

For real numbers, if n is even, the equation y^n = x has a real solution only when x is nonnegative. In that case, there is a unique nonnegative solution, called the principal even root. If x < 0, there is no real root of even order; complex numbers provide roots, which come in n distinct values.

Notation and computation: The general notation is sqrt[n](x) or x^(1/n). The principal root is the nonnegative one for x >= 0. In complex arithmetic, the n roots are spaced evenly around the origin in the complex plane.

Properties: The principal even root is monotone in x for x >= 0, and sqrt[m](sqrt[n](x)) = sqrt[mn](x) for x >= 0. Computation is commonly done by exponentiation or by iterative methods such as Newton–Raphson.

Applications: Even roots arise in geometry, physics, and engineering, including area and normalization calculations, solving polynomial equations, and various normalization procedures.

See also: square root, nth root, radical, principal value, complex roots.