Answer
$i^5=i$
$i^6=-1$
$i^7=-i$
$i^8=1$
$i^9=i$
$i^{10}=-1$
$i^{11}=-i$
$i^{12}=1$
$i^{4k+1}=i$
$i^{4k+2}=-1$
$i^{4k+3}=-i$
$i^{4k}=1$.
To find $i^n$, where $n$ is a positive integer power, find the remainder $r$ of $n\div 4$, you will have $i^n=i^r$.
Work Step by Step
$$i^5=i^4i=1(i)=i$$
$$i^6=i^4i^2=1(-1)=-1$$
$$i^7=i^4i^3=1(-i)=-i$$
$$i^8=i^4i^4=1(1)=1$$
$$i^9=i^8i=1(i)=i$$
$$i^{10}=i^8i^2=1(-1)=-1$$
$$i^{11}=i^{10}i=-1(i)=-i$$
$$i^{12}=i^{10}i^2=-1(-1)=1$$
Notice that the answer repeats every four powers with these ordered values $i$, $-1$, $-i$ and then $1$.
$i^{4k+1}=i$
$i^{4k+2}=-1$
$i^{4k+3}=-i$
$i^{4k}=1$.
To find $i^n$, where $n$ is a positive integer power, find the remainder $r$ of $n\div 4$, you will have $i^n=i^r$.
