  In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following
question: Let $T$ be a subnormal operator, and assume that $T^2$ is
quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this
question in the affirmative. In the present paper, we will extend this result
in two directions. Namely, we prove that both class A $n$th roots of bounded
quasinormal operators and subnormal $n$th roots of unbounded quasinormal
operators are quasinormal. We also show that a non-normal quasinormal operator
having a quasinormal $n$th root has a non-quasinormal $n$th root.
