### Instruction: Express as a common fraction: $(0.\overline{09})(0.\overline{7})$.### Response: 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: $(\frac{a}{b})(\frac{c}{d}) = \frac{ac}{bd}$.