Find the value of $1+i^2+i^4+i^6+i^8+\dots +i^{20}$

If the value $(1+i{)}^6+(1-i{)}^3$ is $a+ib$, then find the value of $a+b$.

Express the complex number $(1+i)(1+2i)$ in the form of $a+ib$.

Express the complex number $\frac{1}{(2+i{)}^2}$ in the standard form of $a+ib$.

Express the complex number $\frac{1-i}{1+i}$ in the standard form of $a+ib$.

If the standard form of the complex number $\frac{(2+i{)}^3}{2+3i}$ is $\frac{m}{13}+\frac{n}{13}i$, then find the value of $m+n$.

Express the complex number $\frac{(1+i)(1+\sqrt{3}i)}{1-i}$ in the standard form of $a+ib$.

Express the complex number $\frac{2+3i}{4+5i}$ in the standard form $a+ib$.