What is the value of `((0.7)^(3) - (0.4)^(3))/((0.7)^(2) + 0.7 xx 0.4 + (0.4)^(2))` ?

Options

A. 0.3
B. 0.4
C. 0.7
D. 1.1

Answer

To solve the expression \(\frac{(0.7)^3 - (0.4)^3}{(0.7)^2 + 0.7 \times 0.4 + (0.4)^2}\), we can use the formula for the difference of cubes and the sum of squares.

### Step 1: Identify the values of A and B
Let \(A = 0.7\) and \(B = 0.4\).

### Step 2: Apply the difference of cubes formula
The formula for the difference of cubes is:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Thus, we can rewrite the numerator:
\[ (0.7)^3 - (0.4)^3 = (0.7 - 0.4)((0.7)^2 + (0.7)(0.4) + (0.4)^2) \]

### Step 3: Calculate \(A - B\)
Calculate \(0.7 - 0.4\):
\[ 0.7 - 0.4 = 0.3 \]

### Step 4: Calculate \(A^2 + AB + B^2\)
Now calculate \(A^2 + AB + B^2\):
\[ (0.7)^2 + (0.7)(0.4) + (0.4)^2 \]
Calculating each term:
- \((0.7)^2 = 0.49\)
- \((0.4)^2 = 0.16\)
- \((0.7)(0.4) = 0.28\)

Now sum them up:
\[ 0.49 + 0.28 + 0.16 = 0.49 + 0.28 = 0.77 \]
\[ 0.77 + 0.16 = 0.93 \]

### Step 5: Substitute back into the expression
Now substitute back into the expression:
\[ \frac{(0.7)^3 - (0.4)^3}{(0.7)^2 + 0.7 \times 0.4 + (0.4)^2} = \frac{(0.3)(0.93)}{0.93} \]

### Step 6: Simplify the expression
Since \(0.93\) in the numerator and denominator cancels out, we have:
\[ \frac{(0.3)(0.93)}{0.93} = 0.3 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0.3} \]