sqrtka2
sqrtka2 is a mathematical notation used to denote the principal square root of the product of a nonnegative constant k and the square of a real variable a, written as sqrt(k a^2). In real arithmetic with k ≥ 0, this expression simplifies to sqrt(k) times the absolute value of a, that is sqrt(k a^2) = sqrt(k) |a|.
- If k ≥ 0 and a is a real number, sqrtka2 is defined as sqrt(k a^2) and yields
- Since a^2 = |a|^2, the expression can be rewritten as sqrt(k a^2) = sqrt(k) |a|, emphasizing its dependence
- Domain: all real numbers a when k ≥ 0. If k < 0, the real-valued square root of
- Range: [0, ∞) for k ≥ 0, since |a| ≥ 0 and sqrt(k) ≥ 0.
- Nonnegativity: sqrtka2 ≥ 0 for k ≥ 0.
- Homogeneity in a: sqrt(k (t a)^2) = |t| sqrt(k a^2) for any real t.
- Lineage to absolute value: sqrt(k a^2) = sqrt(k) |a|, linking the concept to scaling of absolute value.
- If k = 4 and a = -3, sqrtka2 = sqrt(4 * 9) = 6, which equals sqrt(4) * | -3 | = 2 *
- If k = 9 and a = 2, sqrtka2 = sqrt(9 * 4) = 6.
sqrtka2 appears as a compact way to express scaled magnitude in algebra, analysis, and applications where