Prove (i) for nonzero real numbers and integers

Prove (ii) for nonzero real numbers and integers .

Use the definition: for .

23. (i)Proof: We use induction on . Case #1: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for any real numbers and for a non-negative integers . Then . Case #2: . Base case: For , for . Suppose now as induction hypothesis that for some positive integer . Then . Conclusion: Hence, by induction, for any real numbers and for integers .

(ii)Proof:We use induction on . Case #1: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for any real numbers and for non-negative integers . Then . Case #2: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for some positive integer . Then . Case #3: . Base case: For , . (a) or (b) . Inductive step: If (a), then suppose as inductive hypothesis that . Then . Conclusion: Hence, by induction, for any real numbers and for .

For (ii) am I considering the cases correctly? Am I doing this incorrectly?

Thanks