What is the value of `((0.4)^(3)+(0.6)^(3))/([(0.4)^(2)+(0.6)^(2)-(0.4)xx(0.6)]`?

Options

A. a)`1.2`
B. b)`1.1`
C. c)`1.0`
D. d)`0.9`

Answer

To solve the expression \(\frac{(0.4)^3 + (0.6)^3}{(0.4)^2 + (0.6)^2 - (0.4)(0.6)}\), we can use the algebraic identity for the sum of cubes.

### Step 1: Identify the values of A and B
Let \( A = 0.4 \) and \( B = 0.6 \).

### Step 2: Apply the sum of cubes formula
The formula for the sum of cubes is:
\[ A^3 + B^3 = (A + B)(A^2 + B^2 - AB) \]
Thus, we can rewrite the numerator:
\[ (0.4)^3 + (0.6)^3 = (0.4 + 0.6)((0.4)^2 + (0.6)^2 - (0.4)(0.6)) \]

### Step 3: Calculate \( A + B \)
Calculate \( A + B \):
\[ 0.4 + 0.6 = 1.0 \]

### Step 4: Calculate \( A^2 + B^2 \)
Calculate \( (0.4)^2 + (0.6)^2 \):
\[ (0.4)^2 = 0.16, \quad (0.6)^2 = 0.36 \]
So,
\[ (0.4)^2 + (0.6)^2 = 0.16 + 0.36 = 0.52 \]

### Step 5: Calculate \( AB \)
Calculate \( (0.4)(0.6) \):
\[ (0.4)(0.6) = 0.24 \]

### Step 6: Substitute into the formula
Now substitute back into the formula:
\[ (0.4)^2 + (0.6)^2 - (0.4)(0.6) = 0.52 - 0.24 = 0.28 \]

### Step 7: Substitute into the numerator
Now substituting into the numerator:
\[ (0.4)^3 + (0.6)^3 = 1.0 \times 0.28 = 0.28 \]

### Step 8: Substitute into the denominator
The denominator is:
\[ (0.4)^2 + (0.6)^2 - (0.4)(0.6) = 0.28 \]

### Step 9: Final calculation
Now we can simplify the expression:
\[ \frac{(0.4)^3 + (0.6)^3}{(0.4)^2 + (0.6)^2 - (0.4)(0.6)} = \frac{0.28}{0.28} = 1 \]

### Conclusion
The value of the expression is \( \boxed{1} \).