sqrt2a

sqrt2a, written as √(2a), is the principal square root of the product 2a. It denotes the nonnegative number whose square equals 2a. For real numbers a, the expression is defined only when a ≥ 0, since negative radicands do not have real square roots.

In the real numbers, when a ≥ 0, √(2a) satisfies the factorization √(2a) = √2 · √a. This follows from standard radical rules, but it should be applied only to nonnegative a to stay within the real numbers. If a = 0, √(2a) = 0; if a > 0, √(2a) > 0 and grows with the square root of a.

Examples illustrate the idea: a = 8 gives √(16) = 4, and a = 2 gives √(4) = 2. If a can be written as b^2 with b ≥ 0, then √(2a) = √2 · b; more generally, √(2a) = (2a)^(1/2).

In the complex domain, √(2a) requires choosing a branch of the square root. The relation √(xy) = √x · √y is not universally valid without specifying branches, so caution is needed when working with complex values.

Applications of √(2a) appear in contexts where a is scaled by 2 or appears under a square root in formulas, and graphically the function y = √(2a) is a vertically scaled version of y = √a by a factor of √2. See also square root, radical, and exponent 1/2.