What is the value of `((0.3)^(3)-(0.1)^(3))/([(0.3)^(2)+(0.1)^(2)+(0.3)xx (0.1)])`?

Options

A. `0.4`
B. `0.1`
C. `0.5`
D. `0.2`

Answer

To solve the expression \(\frac{(0.3)^3 - (0.1)^3}{(0.3)^2 + (0.1)^2 + (0.3)(0.1)}\), we can use the formula for the difference of cubes and simplify the expression step by step.

### Step 1: Identify A and B
Let \( A = 0.3 \) and \( B = 0.1 \).

### Step 2: Apply the Difference of Cubes Formula
The difference of cubes can be expressed as:
\[ A^3 - B^3 = (A - B)(A^2 + B^2 + AB) \]
So, we can rewrite the numerator:
\[ (0.3)^3 - (0.1)^3 = (0.3 - 0.1)((0.3)^2 + (0.1)^2 + (0.3)(0.1)) \]

### Step 3: Calculate A - B
Calculate \( A - B \):
\[ 0.3 - 0.1 = 0.2 \]

### Step 4: Calculate A² + B² + AB
Now, we need to calculate \( A^2 + B^2 + AB \):
\[ (0.3)^2 = 0.09 \]
\[ (0.1)^2 = 0.01 \]
\[ (0.3)(0.1) = 0.03 \]
Now, add these values together:
\[ 0.09 + 0.01 + 0.03 = 0.13 \]

### Step 5: Substitute Back into the Expression
Now substitute back into the expression:
\[ \frac{(0.3)^3 - (0.1)^3}{(0.3)^2 + (0.1)^2 + (0.3)(0.1)} = \frac{(0.2)(0.13)}{0.13} \]

### Step 6: Simplify the Expression
Now, we can simplify:
\[ \frac{(0.2)(0.13)}{0.13} = 0.2 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0.2} \]