Card-based protocols are cryptographic protocols that perform secure computations using a physical deck of playing cards. In this paper, we focus on card-based protocols that operate with standard commercially available playing cards. A partial-open action is an operation that reveals only a specific portion of a face-down playing card. In 2022, Miyahara and Mizuki proposed a four-card two-input AND protocol using partial-open actions, which requires only a single random cut. In this paper, we extend this line of research by presenting a six-card three-input AND protocol requiring two random cuts and an eight-card four-input AND protocol requiring three random cuts using partial-open actions.