sqrtk9

sqrtk9 is the function f defined by f(k) = sqrt(k^9). It can be rewritten as f(k) = k^(9/2) = k^4 sqrt(k) for k ≥ 0.

The domain of sqrtk9 is k ≥ 0, and the range is f(k) ≥ 0. The function is continuous and strictly increasing on its domain. It is differentiable for k > 0, with derivative f′(k) = (9/2) k^(7/2). At k = 0, the right-hand derivative is 0 in the real-valued setting.

Examples: f(0) = 0, f(1) = 1, f(4) = 512, and f(9) = 19683.

In computations, sqrtk9 is efficiently evaluated as k^4 sqrt(k). For large k, this approach avoids forming k^9 before taking a square root and helps mitigate overflow in some numerical environments. For negative k, the real-valued sqrt is undefined; extending to complex numbers requires choosing a branch for the fractional exponent.

Related topics include power functions, fractional exponents, and the square root function. sqrtk9 illustrates a monomial of degree 9/2 and is relevant to discussions of growth rates and algebraic manipulation involving fractional exponents.