sqrtp3

sqrtp3 is a shorthand for the mathematical expression the square root of p cubed, written as sqrt(p^3) or equivalently p^(3/2). In real arithmetic, this quantity is defined only for nonnegative p, since the square root requires a nonnegative argument.

Definition and interpretation: For real numbers, sqrt(p^3) equals p^(3/2). When p ≥ 0, this can also be written as (sqrt p)^3 or as p * sqrt(p). The identity sqrt(p^3) = p^(3/2) highlights that taking a square root and taking a cube are not interchangeable in general, but for nonnegative p they combine to the same exponent of 3/2.

Domain and limitations: If p < 0, sqrt(p^3) is not a real number because p^3 is negative and the real square root of a negative is undefined. In complex numbers, p^(3/2) can be defined using the complex logarithm, but this introduces branch choices and yields multi-valued results.

Properties: For p ≥ 0, sqrt(p^3) = (sqrt p)^3 = p * sqrt(p) and (sqrt(p^3))^2 = p^3. The function is increasing on [0, ∞). In contexts requiring real values, p must be nonnegative; in broader contexts, a complex interpretation with a chosen branch may be used.

Examples: p = 4 gives sqrt(4^3) = sqrt(64) = 8; p = 1 gives sqrt(1^3) = 1; p = 0 gives 0.

Applications and usage: The expression arises in scaling problems and in formulas where a cube of a parameter is under a square root. The exact meaning depends on the application, and computationally it is often evaluated as p^(3/2) when p is nonnegative.