Answer
$-\dfrac{3}{5}-\dfrac{4}{5}i$
Work Step by Step
Multiplying both the numerator and the denominator of $
\dfrac{i+2}{i-2}
$ by the complex conjugate of the denominator, then
\begin{align*}\require{cancel}
&
\dfrac{i+2}{i-2}\cdot\dfrac{i+2}{i+2}
\\\\&=
\dfrac{i(i)+i(2)+2(i)+2(2)}{(i-2)(i+2)}
&\text{ (use FOIL)}
\\\\&=
\dfrac{i(i)+i(2)+2(i)+2(2)}{(i)^2-(2)^2}
&\text{ (use $(a+b)(a-b)=a^2-b^2$)}
\\\\&=
\dfrac{i^2+2i+2i+4}{i^2-4}
\\\\&=
\dfrac{-1+2i+2i+4}{-1-4}
&\text{ (use $i^2=-1$)}
\\\\&=
\dfrac{(-1+4)+(2i+2i)}{-1-4}
\\\\&=
\dfrac{3+4i}{-5}
\\\\&=
-\dfrac{3}{5}-\dfrac{4}{5}i
.\end{align*}
Hence, the simplified form of the given expression is $
-\dfrac{3}{5}-\dfrac{4}{5}i
$.
