  This paper concerns the exponentiation of monomial ideals. While it is
customary for the exponentiation operation on ideals to consider natural
powers, we extend this notion to powers where the exponent is a positive real
number. Real powers of a monomial ideal generalize the integral closure
operation and highlight many interesting connections to the theory of convex
polytopes. We provide multiple algorithms for computing the real powers of a
monomial ideal. An important result is that given any monomial ideal $I$, the
function taking real numbers to the corresponding real power of $I$ is a step
function whose jumping points are rational. This reduces the problem of
determining real powers to rational exponents.
