Convert the following repeating decimal to a fraction in simplest form. $$ .0 \overline{7} $$

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

1. **Let $$ x $$ be the repeating decimal:**
   $$
   x = 0.77777\ldots
   $$

2. **Multiply both sides by 10 to shift the decimal point:**
   $$
   10x = 7.77777\ldots
   $$

3. **Now, we have two equations:**
   - $$ x = 0.77777\ldots $$
   - $$ 10x = 7.77777\ldots $$

4. **Subtract the first equation from the second:**
   $$
   10x - x = 7.77777\ldots - 0.77777\ldots
   $$
   This simplifies to:
   $$
   9x = 7
   $$

5. **Solve for $$ x $$:**
   $$
   x = \frac{7}{9}
   $$

Thus, the repeating decimal $$ 0.\overline{7} $$ can be expressed as the fraction $$ \frac{7}{9} $$ in simplest form.
$$ 0.\overline{7} = \frac{7}{9} $$

Convert the following repeating decimal to a fraction in simplest form. \[ .0 \overline{7} \]

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