sqrtkm

sqrtkm is a compact notation for the square root of the product of two quantities k and m, written as sqrtkm = sqrt(km). For nonnegative real k and m, this value is real and nonnegative and is commonly interpreted as the geometric mean of k and m.

Domain and extension: If k and m are real numbers with k ≥ 0 and m ≥ 0, sqrtkm is defined in the real numbers. If one or both are negative, the product km may be negative, and the real square root is not defined; in the complex plane, sqrt(km) can be defined as a complex number.

Properties: It is symmetric in k and m: sqrt(km) = sqrt(mk). If k and m are nonnegative and c is nonnegative, sqrt((c k)(c m)) = c sqrt(km). For k, m ≥ 0, min(k, m) ≤ sqrt(km) ≤ max(k, m). This places sqrtkm between the two values and identifies it as the geometric mean when k and m are positive.

Relationship to the geometric mean: The expression sqrtkm equals the geometric mean of k and m for positive k and m.

Computation and examples: Direct evaluation uses sqrt(km). For large numbers, an equivalent form sqrt(km) = exp((ln k + ln m)/2) (with k, m > 0) can help reduce overflow. Edge cases: if either k or m is zero, sqrtkm = 0. Examples: k = 4 and m = 9 give sqrtkm = 6; k = 2 and m = 8 give sqrtkm = 4.

See also: Geometric mean, Square root, Product (mathematics).