To convert the repeating decimal \(0.777...\) (where the 7 repeats indefinitely) into a simplified fraction, you can use the following steps:
Let \( x = 0.777...\).
Now, multiply both sides of the equation by 10:
\[ 10x = 7.777...\]
Next, subtract the original equation from this new equation:
\[ 10x - x = 7.777... - 0.777... \]
This simplifies to:
\[ 9x = 7 \]
Now, solve for \(x\):
\[ x = \frac{7}{9} \]
Thus, the simplified fraction form of the repeating decimal \(0.777...\) is \(\frac{7}{9}\).