Correct multiplication of powers of $i$ I recently noticed a contradiction when multiplying powers of $i$ ($\sqrt {-1}$).
Namely with $i^2 \cdot i^{-1}$.
By resolving the terms first:
$$i^2 \cdot i^{-1} = -1 \cdot -i = i$$
But by combining powers:
$$i^2 \cdot i^{-1} = i^{-2} = \frac{1}{i^2} = \frac{1}{-1} = -1$$
So how should you treat multiplication of powers of imaginary and complex numbers(of the same base)? Is there a set of rules to allow them to be combined, or can they never be combined reliably?
 A: In your second equation, you've written
$$
i^2 \cdot i^{-1} = i^{2 \cdot -1}
$$
(that last thing is $i$ to the power $(2) \cdot (-1)$, by the way)
but the correct formula for 
$$
a^b \cdot a^c
$$
is 
$$
a^ {b+c}.
$$
When you use this, your apparent contradiction disappears. 
But your question is a great one --- you've learned a new definition (powers of complex numbers) and you're checking whether the "rules" you learned for real numbers still work. That's exactly what you should do, and I applaud you for it. 
Let me also point out that in general, even for real numbers, $a^b$ is a little ambiguous: is $4^{\frac{1}{2}}$ equal to $2$ or to $-2$? 
We make a "convention" that it's the first one, but when we try to extend this convention to the complex numbers, it gets problematic, as you'll soon see. So in particular, if you're used to taking an expression like
$$
a^x = b^x
$$
and raising it to the $\frac{1}{x}$ power to get 
$$
(a^x)^{\frac{1}{x}} = (b^x) ^ {\frac{1}{x}}\\
a^{x\cdot{\frac{1}{x}}} = b^{x\cdot {\frac{1}{x}}}\\
a^{1} = b^{1}\\
a = b
$$
then you're going to have to break yourself of that habit. In fact, this doesn't even work for real numbers, as you can see by trying $a = 2, b = -2, x = 2$ -- the problem is that $u \mapsto u^2$ is not a 1-to-1 function, so $u \mapsto u^\frac{1}{2}$ has to involve a choice, and ... things go wrong. 
Roughly speaking: everything like this works find for integer powers, but as soon as you apply it to fractional or irrational powers, things fall apart. 
A: The use of $i$ here is a red herring. Let's replace $i$ with 5.
Then from combining powers we have $5^2 * 5^{-1} = 5^1$, not $5^2 * 5^{-1} = 5^{-2}$.
A: What do you mean by "combining powers"?
Take a non zero complex number $z$ and $n,m\in\Bbb Z$. Then we have that
$$
z^nz^m=z^{n+m}
$$
and NOT that
$$
z^nz^m=z^{nm}
$$
which is what you wrote; this last one is incorrect and leads you to an error in your computation with $z=i$.
