The simplified value of `[(0.111)^(3) + (0.222)^(3) - (0.333)^(3) + (0.333)^(2) (0.222)]^(3)` is :

Options

A. 0.999
B. 0
C. 0.888
D. 0.111

Answer

To simplify the expression \([(0.111)^3 + (0.222)^3 - (0.333)^3 + (0.333)^2 \cdot (0.222)]^3\), we can follow these steps:

### Step 1: Identify the components
Let:
- \( a = 0.111 \)
- \( b = 0.222 \)
- \( c = 0.333 \)

### Step 2: Rewrite the expression
The expression can be rewritten using \( a \), \( b \), and \( c \):
\[ (a^3 + b^3 - c^3 + c^2 \cdot b)^3 \]

### Step 3: Apply the identity for cubes
We know that if \( a + b + c = 0 \), then:
\[ a^3 + b^3 + c^3 = 3abc \]
In our case, we can rewrite \( c \) as \( c = a + b \) (since \( 0.333 = 0.111 + 0.222 \)). Thus:
\[ a + b - c = 0 \]
This allows us to use the identity:
\[ a^3 + b^3 - c^3 = 3abc \]

### Step 4: Calculate \( abc \)
Now we need to calculate \( abc \):
\[ abc = 0.111 \cdot 0.222 \cdot 0.333 \]

Calculating \( abc \):
\[ 0.111 \cdot 0.222 = 0.024642 \]
\[ 0.024642 \cdot 0.333 = 0.008206 \]

### Step 5: Substitute back into the identity
Now substituting back into the identity:
\[ a^3 + b^3 - c^3 = 3 \cdot 0.008206 = 0.024618 \]

### Step 6: Add \( c^2 \cdot b \)
Next, we need to calculate \( c^2 \cdot b \):
\[ c^2 = (0.333)^2 = 0.110889 \]
\[ c^2 \cdot b = 0.110889 \cdot 0.222 = 0.024618 \]

### Step 7: Combine the results
Now we can combine the results:
\[ a^3 + b^3 - c^3 + c^2 \cdot b = 0.024618 + 0.024618 = 0.049236 \]

### Step 8: Cube the result
Finally, we need to cube the result:
\[ (0.049236)^3 \]

Calculating \( (0.049236)^3 \):
\[ 0.049236^3 \approx 0.000119 \]

### Final Answer
Thus, the simplified value of the original expression is approximately:
\[ \boxed{0.000119} \]