Answer
$1-i$.
Work Step by Step
The given expression is
$=\frac{i^4+i^{12}}{i^8-i^7}$
Factor each term.
$=\frac{(i^2)^2+(i^{2})^6}{(i^2)^4-(i^2)^3i}$
Use $i^2=-1$
$=\frac{(-1)^2+(-1)^6}{(-1)^4-(-1)^3i}$
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.
$=\frac{2(1-i)}{2}$
$=1-i$.
