Answer
$1+i$.
Work Step by Step
The given expression is
$=\frac{i^8+i^{40}}{i^4+i^3}$
Factor each term.
$=\frac{(i^2)^4+(i^{2})^{20}}{(i^2)^2+i^2i}$
Use $i^2=-1$
$=\frac{(-1)^4+(-1)^{20}}{(-1)^2+(-1)i}$
Simplify.
$=\frac{1+1}{1-i}$
$=\frac{2}{1-i}$
The conjugate of the denominator is $1+i$.
Multiply the numerator and the denominator by $1+i$.
$=\frac{2}{1-i}\cdot \frac{1+i}{1+i}$
Use the special formula $(A+B)^2=A^2+2AB+B^2$
$=\frac{2(1+i)}{1^2-i^2}$
Use $i^2=-1$.
$=\frac{2(1+i)}{1+1}$
Simplify.
$=\frac{2(1+i)}{2}$
$=1+i$.
