Answer
$0$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
i^7+i^5+i^3+i
,$ use the laws of exponents and the equivalence $i^2=-1.$
$\bf{\text{Solution Details:}}$
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
i^{6+1}+i^{4+1}+i^{2+1}+i
\\\\=
i^6\cdot i^1+i^4\cdot i^1+i^{2}\cdot i^1+i
\\\\=
i^6\cdot i+i^4\cdot i+i^{2}\cdot i+i
.\end{array}
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}\cdot i+i^{2\cdot2}\cdot i+i^{2}\cdot i+i
\\\\
(i^2)^3\cdot i+(i^2)^2\cdot i+i^{2}\cdot i+i
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(-1)^3\cdot i+(-1)^2\cdot i+(-1)\cdot i+i
\\\\=
(-1)\cdot i+(1)\cdot i+(-1)\cdot i+i
\\\\=
-i+i-i+i
\\\\=
0
.\end{array}
