To convert the repeating decimal $0.\overline{7}$ to a simplified fraction, we can follow these steps:

1. Let $x = 0.\overline{7}$.
2. Multiply both sides by 10 to shift the decimal point one place to the right: $10x = 7.\overline{7}$.
3. Subtract the original equation from this new equation to eliminate the repeating part: $10x - x = 7.\overline{7} - 0.\overline{7}$ $9x = 7$
4. Solve for $x$: $x = \frac{7}{9}$

Thus, $0.\overline{7}$ as a simplified fraction is $\frac{7}{9}$.

Would you like more details or have any questions?

Here are 5 more questions you can ask:

1. How do you convert a repeating decimal like $0.\overline{123}$ to a fraction?
2. Can you explain how to solve a quadratic equation?
3. What is the process for finding the derivative of a function?
4. How do you integrate a trigonometric function?
5. Can you help with simplifying complex fractions?

Tip: When converting repeating decimals to fractions, always start by defining the repeating decimal as a variable and then use algebraic manipulation to eliminate the repeating part.