$A$ and $B$ are integers such that $A^B = 4$ . Does this mean that $A\times B= 4?$

Yes
No

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$A \times B = -4$ is also true. If $A = -2$ and $B =2$ then $A^B = (-2)^2 = 4,$ so $A \times B = -2 \times 2 = -4.$