jmc

We begin by expressing $0.\overline{09}$ and $0.\overline{7}$ as common fractions.

To express the number $0.\overline{09}$ as a fraction, we call it $x$ and subtract it from $100x$: \begin{array}{r r c r@{}l} &100x &=& 9&.090909\ldots \\ - &x &=& 0&.090909\ldots \\ \hline &99x &=& 9 & \end{array} This shows that $0.\overline{09}=\frac{9}{99}=\frac{1}{11}$.

We can do a similar trick to express $0.\overline{7}$ as a fraction. Call this number $y$ and subtract it from $10y$: \begin{array}{r r c r@{}l} &10y &=& 7&.77777\ldots \\ - &y &=& 0&.77777\ldots \\ \hline &9y &=& 7 & \end{array} This shows that $0.\overline{7}=\frac{7}{9}$.

Therefore, $(0.\overline{09})(0.\overline{7})=\frac{1}{11}\cdot \frac{7}{9}=\boxed{\frac{7}{99}}$.