Answer
$i^{4446}=$-1
Work Step by Step
We know that any number to the power of zero is 1, so $i^0=1$
We also know that $i^2=-1$, so
$i^3=i^2 \cdot i^1=-1 \cdot i=-i$
and
$i^4=i^2\cdot i^2=-1\cdot -1=1$
Using this iteration process, we'll can find that:
$i^5=i^4\cdot i^1=1\cdot i=i$
$i^6=i^4\cdot i^2=1\cdot -1=-1$
$i^7=i^4\cdot i^3=1\cdot -i=-i$
$i^8=i^4\cdot i^4=1\cdot 1=1$
$i^9=i^4\cdot i^4 \cdot i^1=1\cdot 1 \cdot i=i$
$i^{10}=i^4\cdot i^4 \cdot i^2=1\cdot 1\cdot -1=-1$
$i^{11}=i^4\cdot i^4 \cdot i^3=1\cdot 1\cdot -i=-i$
$i^{12}=i^4\cdot i^4 \cdot i^4=1\cdot 1\cdot 1=1$
Putting them all up in order, we see that the pattern $1, i, -1, -i$ repeats itself:
$i^0=1$
$i^1=i$
$i^2=-1$
$i^3=-i$
$i^4=1$
$i^5=i$
$i^6=-1$
$i^7=-1$
$i^8=1$
$i^9=i$
$i^{10}=-1$
$i^{11}=-1$
$i^{12}=1$
We can see in the iteration process that any power of $i$ can be simplified to $i^0, i^1, i^2,$ or $i^3$ by reducing the exponent by 4 as many times as needed since $i^4 =1$ and all numbers multiplied by $1$ gives that same number. Therefore to calculate any power of $i$, we'll need to divide the exponent by 4 and find the remainder. The solution will be $i$ to the power of the remainder. It works well because the remainder can only be 0, 1, 2 or 3.
Now we can calculate $i^{4446}$. First, we divide 4446 by 4, giving us 1111 and a remainder of 2. So, the answer is $i^2$ which is $-1$.
