Calling All Powers That Give Two Digits

Number Theory Level 2

Let $A,B,C$ and $D$ be distinct single digit positive integers such that: $A^B =\overline{CD} \text{ and } A+B=C+D$

Compute $\overline{AB} + \overline{BC} + \overline{CD} + \overline{DA}$ .

Clarification: $\overline{AB}$ denotes a 2-digit integer with $A$ and $B$ as its digits.