TITLE: Basic Mathematics. Trouble with proof, powers and odd numbers.
QUESTION [10 upvotes]: Greets,
In the exercises, at the end of chapter 1.4, Basic Mathematics, Serge Lang
6) Prove: If $n$ is odd, then $\quad (-1)^n = -1$
How?
The working I did
$$\begin{align}( -1)^n &= ( -1 )^{2m+1}\\
  &= ( ( -1)^2 )^{m+1}\\
  &= ( ( -1 )( -1 ) )^{m+1}\\
  &= ( 1 )^{m+1} \end{align}$$
Can it get from here to $-1$?
The book shows
Let  $n =  2k + 1$. Then
$$\begin{align}(-1)^n  &= (-1)^{2k+1}\\
   &= (-1)^{2k}(-1)\\
   &= 1 \cdot (-1)\\
   &= -1 \end{align}$$
I'm having trouble with how $(-1)^{2k+1}$ became $(-1)^{2k}(-1)$.

REPLY [3 votes]: You made an error in the 2nd equality. The relevant property of exponentiation here is:
$$
a^{m+n}=a^m a^n
$$