Answer
$0$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
i^6+i^4+i^2+1
,$ use the laws of exponents and the equivalence $i^2=-1.$
$\bf{\text{Solution Details:}}$
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
i^{2\cdot3}+i^{2\cdot2}+i^2+1
\\\\=
(i^2)^3+(i^2)^2+i^2+1
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(-1)^3+(-1)^2+(-1)+1
\\\\=
-1+1-1+1
\\\\=
0
.\end{array}
