The value of `((0.67xx0.67xx0.67)-(0.33xx0.33xx0.33))/((0.67xx0.67)+(0.67xx0.33)+(0.33xx0.33))` is

Options

A. `11`
B. `1.1`
C. `3.4`
D. `0.34`

Answer

To solve the expression \(\frac{(0.67 \times 0.67 \times 0.67) - (0.33 \times 0.33 \times 0.33)}{(0.67 \times 0.67) + (0.67 \times 0.33) + (0.33 \times 0.33)}\), we can follow these steps:

### Step 1: Identify the components
We can rewrite the expression using powers:
- The numerator becomes \(0.67^3 - 0.33^3\).
- The denominator can be rewritten as \(0.67^2 + 0.67 \times 0.33 + 0.33^2\).

### Step 2: Apply the difference of cubes formula
The difference of cubes formula states that:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Here, let \(A = 0.67\) and \(B = 0.33\). Therefore, we can express the numerator as:
\[ (0.67 - 0.33)(0.67^2 + 0.67 \times 0.33 + 0.33^2) \]

### Step 3: Calculate \(A - B\)
Calculate \(0.67 - 0.33\):
\[ 0.67 - 0.33 = 0.34 \]

### Step 4: Calculate the denominator
The denominator is already in the form of \(A^2 + AB + B^2\):
\[ 0.67^2 + 0.67 \times 0.33 + 0.33^2 \]
Calculating each term:
- \(0.67^2 = 0.4489\)
- \(0.33^2 = 0.1089\)
- \(0.67 \times 0.33 = 0.2211\)

Now, add these values:
\[ 0.4489 + 0.2211 + 0.1089 = 0.7789 \]

### Step 5: Substitute back into the expression
Now substitute back into the expression:
\[ \frac{0.34 \times (0.67^2 + 0.67 \times 0.33 + 0.33^2)}{0.7789} \]
Since the numerator simplifies to \(0.34 \times 0.7789\), we can cancel out \(0.7789\):
\[ \frac{0.34 \times 0.7789}{0.7789} = 0.34 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0.34} \]