sqrtxn

sqrtxn is a compact notation used to denote the square root of the product x times n, written as sqrt(xn). In contexts where x and n are real numbers and the product xn is nonnegative, sqrtxn refers to the principal nonnegative square root of that product. If xn is negative in the real numbers, the real-valued square root is undefined; in the complex numbers the square root is defined with a chosen branch, and the value can depend on that choice.

Properties of sqrtxn follow standard radical rules. For nonnegative x and n, sqrt(xn) = sqrt(x) sqrt(n). This identity is commonly used to simplify expressions and solve equations. In the complex setting, the product rule for square roots can fail if branches are not chosen consistently, so one must be careful when extending to complex numbers.

Examples help illustrate the concept. If x = 2 and n = 3, sqrtxn = sqrt(6) ≈ 2.449. If x = 4 and n = 9, sqrtxn = sqrt(36) = 6. These demonstrate how the square root of a product can reduce to a root of the product or separate into the product of roots when appropriate.

Notation notes are useful to avoid ambiguity. sqrtxn is typically written with parentheses as sqrt(xn) to indicate the argument is the entire product, not a product of separate square roots unless explicitly stated. In real-number contexts with nonnegative factors, sqrt(xn) can be expressed as sqrt(x) sqrt(n).