sqrtab

sqrtab is a shorthand associated with the square root of a product, commonly written as sqrt(ab). It denotes the nonnegative principal square root of the product a times b, usually in the real numbers, provided that the product ab is nonnegative.

In real-number arithmetic, there is a convenient property: if a ≥ 0 and b ≥ 0, then sqrt(ab) = sqrt(a) sqrt(b). This multiplicative rule relies on the nonnegativity of the factors. If ab ≥ 0 but either a or b is negative, the expression sqrt(ab) is real only when both are nonnegative; if both are negative, ab is positive but sqrt(a) and sqrt(b) are not real, so the simple product rule does not apply within the real numbers. When complex numbers are allowed, the relationship between sqrt(ab) and sqrt(a) sqrt(b) depends on branch choices of the square-root function and is not universally valid without additional conventions.

Key identities include sqrt(x^2) = |x| for real x, and sqrt(uv) ≥ 0 for real u, v with uv ≥ 0. The square root function is defined as (ab)^{1/2} in exponent form, which highlights its behavior under multiplication and exponent rules, with caveats noted above in nonalk real or complex contexts.

Examples: sqrt(12) equals sqrt(4*3) = 2 sqrt(3); sqrt(18) equals 3 sqrt(2). If a = 4 and b = 9, sqrt(ab) = 6. When programming or in certain mathematical contexts, sqrtab may be used to denote sqrt(a*b) to emphasize that the product is taken before taking the square root.

Applications of sqrtab appear in algebra, geometry, and calculus, where simplifying radicals and solving equations often rely on the property sqrt(ab) = sqrt(a) sqrt(b) under appropriate conditions.