Express `0.bar(7)` in the simplest form.

Answer

To express \(0.\overline{7}\) in its simplest form, we can follow these steps:

### Step 1: Let \(x\) be the repeating decimal
Let:
\[ x = 0.\overline{7} \]

### Step 2: Multiply by 10
To eliminate the repeating part, multiply both sides of the equation by 10:
\[ 10x = 7.\overline{7} \]

### Step 3: Set up the equation
Now, we have two equations:
1. \(x = 0.\overline{7}\)
2. \(10x = 7.\overline{7}\)

### Step 4: Subtract the first equation from the second
Subtract the first equation from the second:
\[ 10x - x = 7.\overline{7} - 0.\overline{7} \]
This simplifies to:
\[ 9x = 7 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 9 to solve for \(x\):
\[ x = \frac{7}{9} \]

### Conclusion
Thus, the simplest form of \(0.\overline{7}\) is:
\[ \frac{7}{9} \]

---