  In this work we derived a rotating and non-linear magnetic-charged black hole
surrounded by quintessence using the Newman-Janis algorithm. Considering the
state parameter $\omega_q=-3/2$, we studied the event horizons, the ergosphere,
and the ZAMO. We found that the existence of the outer horizon is constrained
by the values of the charge $Q$. Furthermore, we found that the ergo-region
increases when both the charge $Q$ and the spin parameter $a$ are increased. On
the other hand, regarding equatorial circular orbits, we studied the limit
given by the static radius on the existence of circular geodesics, the photon
circular geodesics, and the innermost stable circular orbits (ISCO). We show
that photon circular orbits do not depend strongly on $Q$, and $r_{ISCO}$ is
constrained by the values of charge.
