In this paper, we consider the family of rational maps $$\F(z) = z^n +
\frac{\la}{z^d},$$ where $n \geq 2$, $d\geq 1$, and$\la \in \bbC$. We consider
the case where $\la$ lies in the main cardioid of one of the $n-1$ principal
Mandelbrot sets in these families. We show that the Julia sets of these maps
are always homeomorphic. However, two such maps $\F$ and $F_\mu$ are conjugate
on these Julia sets only if the parameters at the centers of the given
cardioids satisfy $\mu = \nu^{j(d+1)}\la$ or $\mu = \nu^{j(d+1)}\bar{\la}$
where $j \in \bbZ$ and $\nu$ is an $n-1^{\rm st}$ root of unity.