$(A[1])^{\otimes n}\backsimeq (A^{\otimes n})[n]$?

Question

When $A$ is a $\mathbb{Z}$-graded module, $A[1]$ is the shift or suspension of $A$ (i.e $(A[1])^{i}=A^{i+1}$). May the $n$th power tensor of the shift be identified in this way?. Am I missing anything?. What if $A$ is a differential graded algebra?.