Answer
$\dfrac{2(a+2)}{5a}$
Work Step by Step
The given expression, $
\dfrac{a^{-1}+2a^{-2}}{2a^{-1}+(2a)^{-1}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\dfrac{1}{a}+\dfrac{2}{a^2}}{\dfrac{2}{a}+\dfrac{1}{2a}}
\\\\=
\dfrac{\dfrac{a+2}{a^2}}{\dfrac{4+1}{2a}}
\\\\=
\dfrac{\dfrac{a+2}{a^2}}{\dfrac{5}{2a}}
\\\\=
\dfrac{a+2}{a^2}\div\dfrac{5}{2a}
\\\\=
\dfrac{a+2}{a^2}\cdot\dfrac{2a}{5}
\\\\=
\dfrac{a+2}{\cancel{a}\cdot a}\cdot\dfrac{2\cancel{a}}{5}
\\\\=
\dfrac{2(a+2)}{5a}
.\end{array}
