# Prove conjugate of imaginary number

This is statement is used in one of the proofs in my book and I am struggling to figure out how to deduce it or proof it:

$, \overline{i^{n - k}} = i^{n - k}(-1)^{n - k})$

What is the intuition behind the statement? Why is the $(-1)$ raised to a power?

Wouldn't the conjugate of $i$ be $-i$ and thus the conjugate of $i^n$ be $-i^n$?

• $(-i)^n$, not $-i^n$. Sep 28, 2015 at 13:45

Simply, we have $\overline{zz'}= \bar{z}\bar{z'}$. So $$\overline{i^{n-k}}=\bar{i}\bar{i}...\bar{i}= (-i)(-i)....(-i)= (-1)^{n-k} \; \;i^{n-k}$$
$$i=e^{i\frac{\pi}{2}},$$
and discuss the cases where $n-k\in \{4m, 4m+1,4m +2,4m+3\}.$