The simplification of `(0.bar1)^2{1-9(0.1bar6)^2}` is

Options

A. `(-1)/162`
B. `1/108`
C. `7696/(10^6)`
D. `1/109`

Answer

To simplify the expression \( (0.\overline{1})^2 \{ 1 - 9(0.1\overline{6})^2 \} \), we will follow these steps:

### Step 1: Convert repeating decimals to fractions
First, we convert \( 0.\overline{1} \) and \( 0.1\overline{6} \) into fractions.

- For \( 0.\overline{1} \):
Let \( x = 0.\overline{1} \).
Then, \( 10x = 1.\overline{1} \).
Subtracting these two equations:
\[ 10x - x = 1.\overline{1} - 0.\overline{1} \implies 9x = 1 \implies x = \frac{1}{9} \]
Thus, \( 0.\overline{1} = \frac{1}{9} \).

- For \( 0.1\overline{6} \):
Let \( y = 0.1\overline{6} \).
Then, \( 100y = 16.\overline{6} \).
Subtracting these two equations:
\[ 100y - y = 16.\overline{6} - 0.1\overline{6} \implies 99y = 16.5 \implies y = \frac{16.5}{99} = \frac{165}{990} = \frac{11}{66} = \frac{1}{6} \]
Thus, \( 0.1\overline{6} = \frac{1}{6} \).

### Step 2: Substitute the fractions into the expression
Now, we substitute these fractions back into the original expression:
\[ (0.\overline{1})^2 = \left(\frac{1}{9}\right)^2 = \frac{1}{81} \]
\[ (0.1\overline{6})^2 = \left(\frac{1}{6}\right)^2 = \frac{1}{36} \]
Now substituting these into the expression:
\[ \frac{1}{81} \left( 1 - 9 \cdot \frac{1}{36} \right) \]

### Step 3: Simplify the expression inside the brackets
Calculating \( 9 \cdot \frac{1}{36} \):
\[ 9 \cdot \frac{1}{36} = \frac{9}{36} = \frac{1}{4} \]
Now substituting this back:
\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Combine the results
Now, we have:
\[ \frac{1}{81} \cdot \frac{3}{4} = \frac{3}{324} = \frac{1}{108} \]

### Final Answer
Thus, the simplification of the expression \( (0.\overline{1})^2 \{ 1 - 9(0.1\overline{6})^2 \} \) is:
\[ \frac{1}{108} \]