What is the value of `((0.5)^(3)-(0.1)^(3))/((0.5)^(2)+0.5xx0.1+(0.1)^(2))` ?

Options

A. a) `0.1`
B. b) `0.4`
C. c) `0.5`
D. d) `0.6`

Answer

To solve the expression \(\frac{(0.5)^3 - (0.1)^3}{(0.5)^2 + 0.5 \times 0.1 + (0.1)^2}\), we can use the formula for the difference of cubes and the identity for the sum of squares.

### Step-by-Step Solution:

1. **Identify A and B**:
Let \( A = 0.5 \) and \( B = 0.1 \).

2. **Apply the Difference of Cubes Formula**:
The difference of cubes can be expressed as:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Therefore, we can rewrite the numerator:
\[ (0.5)^3 - (0.1)^3 = (0.5 - 0.1)((0.5)^2 + (0.5)(0.1) + (0.1)^2) \]

3. **Calculate \( A - B \)**:
\[ A - B = 0.5 - 0.1 = 0.4 \]

4. **Calculate \( A^2 + AB + B^2 \)**:
- First, calculate \( A^2 \):
\[ (0.5)^2 = 0.25 \]
- Next, calculate \( AB \):
\[ 0.5 \times 0.1 = 0.05 \]
- Finally, calculate \( B^2 \):
\[ (0.1)^2 = 0.01 \]
- Now, sum these values:
\[ 0.25 + 0.05 + 0.01 = 0.31 \]

5. **Substitute Back into the Expression**:
Now substituting back into the expression:
\[ \frac{(0.5)^3 - (0.1)^3}{(0.5)^2 + 0.5 \times 0.1 + (0.1)^2} = \frac{(0.4)(0.31)}{0.31} \]

6. **Simplify the Expression**:
Since \( 0.31 \) in the numerator and denominator cancels out:
\[ = 0.4 \]

### Final Answer:
The value of the expression is \( 0.4 \).