Express as a common fraction: $(0.\overline{09})(0.\overline{7})$.
I notice that both factors are repeating decimals, which means they can be written as fractions using a clever trick. For example, to write $0.\overline{09}$ as a fraction, I can let $x = 0.\overline{09}$ and multiply both sides by 100 to get $100x = 9.\overline{09}$. Then I can subtract $x$ from $100x$ to eliminate the repeating part and get $99x = 9$. Dividing both sides by 99, I get $x = \frac{9}{99}$, which can be simplified to $x = \frac{1}{11}$. Similarly, to write $0.\overline{7}$ as a fraction, I can let $y = 0.\overline{7}$ and multiply both sides by 10 to get $10y = 7.\overline{7}$. Then I can subtract $y$ from $10y$ and get $9y = 7$. Dividing both sides by 9, I get $y = \frac{7}{9}$. Now that I have both factors as fractions, I can multiply them using the rule for multiplying fractions: $(\frac{a}{b})(\frac{c}{d}) = \frac{ac}{bd}$. So, $(0.\overline{09})(0.\overline{7}) = (\frac{1}{11})(\frac{7}{9}) = \frac{7}{99}$.

