sqrt2k

sqrt2k denotes the square root of the product 2k, usually written as sqrt(2k). It is a real number for k greater than or equal to zero, and can be extended to complex values when k is negative, by using the complex square root.

If k is an integer, sqrt(2k) is irrational unless 2k is a perfect square. This occurs precisely when k = 2m^2 for some integer m, in which case sqrt(2k) = 2m. In general, sqrt(2k) cannot be simplified to a rational number unless 2k is a square.

Basic properties include that sqrt(2k) is nonnegative for k ≥ 0 and that, for nonnegative k, sqrt(2k) = sqrt(2) sqrt(k). The function is monotone increasing in k on the nonnegative integers and reals, growing roughly like the square root of k.

Examples illustrate the values: k = 1 gives sqrt(2) ≈ 1.4142; k = 2 gives sqrt(4) = 2; k = 3 gives sqrt(6) ≈ 2.4495; k = 5 gives sqrt(10) ≈ 3.1623.

Applications and context: sqrt(2k) appears as a scaling factor in geometric, probabilistic, and number-theoretic problems, including estimates involving distances, areas, or lattice-point counts on circles of radius sqrt(2k). It also serves as a simple benchmark for numerical approximation and for understanding when such an expression simplifies to an integer.