I am currently working in my Discrete math class with elementary number theory and methods of proof. I have been given the problem $-a^n = (-a)^n$. According to the professor and the book this property is true for some integers and false for others integers. For example: Let $a=1$. Then, $-1^n$ = $(-1)^n$. Wouldn't that be true? But How can I show an example where this property is false for other integers?
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Yes, let a = 1. Then we can NOT say $-1^n = (-1)^n$ for all n. For example take n = 2. Well, $-1^2 = -1$, but $(-1)^2 = 1 \neq -1$. Now consider n = 3. This time it works: $-1^3 = (-1)^3 = -1$. Therefore have shown that $-a^n = (-a)^n$ only holds for some combinations of a and n.
To be more precise we can say that the equation only holds for odd $n$ (assuming n is an integer, to avoid imaginary numbers). It turns out that whether this equation holds is actually independent of a. Choose any $a$ you'd like and the equation will only hold for odd $n$.