Question:
Let $a = .\overline{2} + .\overline{6}$.  Find the reciprocal of $a$, expressed as a decimal.

Answer:
Convert both decimals into fractions. \begin{align*}
x&=.\overline{2} \\
\Rightarrow 10x&=2.\overline{2} \\
\Rightarrow 9x&=2 \\
\Rightarrow x &= \frac{2}{9}.
\end{align*}Likewise, $.\overline{6}=\frac{6}{9}$. Adding the two fractions yields $\frac{2}{9} + \frac{6}{9}=\frac{8}{9}$. The reciprocal of this is $\frac{1}{\frac{8}{9}}=\frac{9}{8}$. To convert this to a decimal, we need to multiply the numerator and denominator by 125. Doing so, we get \[\frac{9}{8} \cdot \frac{125}{125} = \frac{9 \cdot 125}{8 \cdot 125} = \frac{1125}{1000}=\boxed{1.125}.\]