  If $a$ and $b$ are a pair of invertible elements, then $ab$ is also
invertible and the inverse of the product $ab$ satisfying
  $$(ab)^{-1}=a^{-1}b^{-1}$$ is known as the {\it forward-order law}.
  This article establishes a few sufficient conditions of the forward-order law
for the core inverse of elements in rings with involution. It also presents the
forward-order law for the weighted core inverse and the triple forward-order
law for the core inverse. Additionally, we discuss the hybrid forward-order law
among the Moore-Penrose inverse, the group inverse, and the core inverse.
