Let \( x \) be the repeating decimal \( 0.\overline{8} \).
This means that:
\[ x = 0.88888\ldots \]
To convert this repeating decimal to a fraction, we can set up the following equation:
\[ 10x = 8.88888\ldots \]
Now, we have two equations:
- \( x = 0.88888\ldots \)
- \( 10x = 8.88888\ldots \)
Next, we can subtract the first equation from the second:
\[ 10x - x = 8.88888\ldots - 0.88888\ldots \]
This simplifies to:
\[ 9x = 8 \]
Now, divide both sides by 9:
\[ x = \frac{8}{9} \]
Thus, the repeating decimal \( 0.\overline{8} \) is equal to the fraction \( \frac{8}{9} \).
To ensure this is in simplest form, we note that 8 and 9 have no common factors other than 1. Therefore, the fraction \( \frac{8}{9} \) is already in simplest form.
The final answer is:
\[ \boxed{\frac{8}{9}} \]