[tex]\bf \cfrac{8-5i}{5+8i}\cdot \cfrac{5-8i}{5-8i}\impliedby \textit{using the conjugate of the denominator}
\\\\\\
\textit{now recall }\textit{difference of squares}
\\ \quad \\
(a-b)(a+b) = a^2-b^2\qquad \qquad
a^2-b^2 = (a-b)(a+b)\\\\
-----------------------------\\\\[/tex]
[tex]\bf \cfrac{(8-5i)(5-8i)}{(5+8i)(5-8i)}\implies \cfrac{40-64i-25i+40i^2}{5^2-(8i)^2}\\\\
-----------------------------\\\\
\textit{now, also recall }i^2=-1\\\\
-----------------------------\\\\
\cfrac{40-89i+40(-1)}{25-(8^2i^2)}\implies \cfrac{40-40-89i}{25-64(-1)}\implies \cfrac{-89i}{25+64}
\\\\\\
\cfrac{-89i}{89}\implies -i\iff 0-i[/tex]
