What is the value of:
`((0.101)_(2)^((11)_(2))+(0.011)_(2)^((11)_(2)))/((0.101)_(2)^((10)_(2))-(0.101)_(2)^((01)_(2))(0.011)_(2)^((01)_(2))+(0.011)_(2)^((10)_(2)))`

Options

A. `(0.001)_(2)`
B. `(0.01)_(2)`
C. `(0.1)_(2)`
D. `(1)_(2)`

Answer

To solve the given expression step by step, we will first convert the binary numbers to decimal, perform the arithmetic operations, and then convert the final result back to binary.

### Step 1: Convert binary fractions to decimal
1. **Convert \( (0.101)_2 \) to decimal:**
\[ 0.101_2 = 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = \frac{1}{2} + 0 + \frac{1}{8} = \frac{4}{8} + \frac{1}{8} = \frac{5}{8} \]

2. **Convert \( (0.011)_2 \) to decimal:**
\[ 0.011_2 = 0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3} = 0 + \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]

3. **Convert \( (11)_2 \) to decimal:**
\[ (11)_2 = 1 \times 2^1 + 1 \times 2^0 = 2 + 1 = 3 \]

4. **Convert \( (10)_2 \) to decimal:**
\[ (10)_2 = 1 \times 2^1 + 0 \times 2^0 = 2 + 0 = 2 \]

5. **Convert \( (01)_2 \) to decimal:**
\[ (01)_2 = 0 \times 2^1 + 1 \times 2^0 = 0 + 1 = 1 \]

### Step 2: Substitute the decimal values into the expression
Now, we can substitute these values into the original expression:
\[ \frac{(0.101)_2^{(11)_2} + (0.011)_2^{(11)_2}}{(0.101)_2^{(10)_2} - (0.101)_2^{(01)_2} \cdot (0.011)_2^{(01)_2} + (0.011)_2^{(10)_2}} \]

Substituting the decimal equivalents:
\[ \frac{\left(\frac{5}{8}\right)^3 + \left(\frac{3}{8}\right)^3}{\left(\frac{5}{8}\right)^2 - \left(\frac{5}{8}\right)^1 \cdot \left(\frac{3}{8}\right)^1 + \left(\frac{3}{8}\right)^2} \]

### Step 3: Calculate the numerator and denominator
1. **Calculate the numerator:**
\[ \left(\frac{5}{8}\right)^3 + \left(\frac{3}{8}\right)^3 = \frac{125}{512} + \frac{27}{512} = \frac{152}{512} \]

2. **Calculate the denominator:**
\[ \left(\frac{5}{8}\right)^2 - \left(\frac{5}{8}\right) \cdot \left(\frac{3}{8}\right) + \left(\frac{3}{8}\right)^2 \]
\[ = \frac{25}{64} - \frac{15}{64} + \frac{9}{64} = \frac{25 - 15 + 9}{64} = \frac{19}{64} \]

### Step 4: Final calculation
Now we can put the numerator and denominator together:
\[ \frac{\frac{152}{512}}{\frac{19}{64}} = \frac{152}{512} \cdot \frac{64}{19} = \frac{152 \cdot 64}{512 \cdot 19} \]
\[ = \frac{9728}{9728} = 1 \]

### Step 5: Convert the result back to binary
The decimal value \( 1 \) in binary is:
\[ (1)_2 \]

### Final Answer
The value of the expression is \( (1)_2 \).