What is the Value of `(0.74xx1.23xx0.13)/((0.37)^3+(0.41)^3-8(0.39)^3)`?

Options

A. `(-1)/3`
B. 1
C. `-1`
D. `1/3`

Answer

To solve the expression \((0.74 \times 1.23 \times 0.13) / ((0.37)^3 + (0.41)^3 - 8(0.39)^3)\), we can follow these steps:

### Step 1: Rewrite the Numerator
We start with the numerator \(0.74 \times 1.23 \times 0.13\). We can express \(0.74\) and \(1.23\) in terms of \(0.37\) and \(0.41\):
\[ 0.74 = 2 \times 0.37 \quad \text{and} \quad 1.23 = 3 \times 0.41 \]
Thus, we can rewrite the numerator as:
\[ (2 \times 0.37) \times (3 \times 0.41) \times 0.13 = 6 \times 0.37 \times 0.41 \times 0.13 \]

### Step 2: Rewrite the Denominator
Next, we simplify the denominator:
\[ (0.37)^3 + (0.41)^3 - 8(0.39)^3 \]
Using the identity \(a^3 + b^3 - 3ab(a + b)\) for \(a = 0.37\), \(b = 0.41\), and \(c = 0.39\):
\[ 0.37^3 + 0.41^3 = (0.37 + 0.41)(0.37^2 - 0.37 \cdot 0.41 + 0.41^2) \]
Calculating \(0.37 + 0.41 = 0.78\).

Now, we need to calculate \(0.37^2 - 0.37 \cdot 0.41 + 0.41^2\):
\[ 0.37^2 = 0.1369, \quad 0.41^2 = 0.1681, \quad 0.37 \cdot 0.41 = 0.1517 \]
Thus,
\[ 0.37^2 - 0.37 \cdot 0.41 + 0.41^2 = 0.1369 - 0.1517 + 0.1681 = 0.1533 \]
So,
\[ 0.37^3 + 0.41^3 = 0.78 \times 0.1533 \]

### Step 3: Calculate \(8(0.39)^3\)
Now, we calculate \(8(0.39)^3\):
\[ (0.39)^3 = 0.059319 \quad \Rightarrow \quad 8(0.39)^3 = 8 \times 0.059319 = 0.474552 \]

### Step 4: Combine the Denominator
Now, we can combine the denominator:
\[ 0.37^3 + 0.41^3 - 8(0.39)^3 = (0.78 \times 0.1533) - 0.474552 \]
Calculating \(0.78 \times 0.1533 = 0.119574\):
\[ 0.119574 - 0.474552 = -0.354978 \]

### Step 5: Final Calculation
Now we can substitute back into our original expression:
\[ \frac{6 \times 0.37 \times 0.41 \times 0.13}{-0.354978} \]

### Step 6: Simplify
Calculating \(6 \times 0.37 \times 0.41 \times 0.13\):
\[ 6 \times 0.37 \times 0.41 \times 0.13 = 0.0366 \]
Thus, the final value is:
\[ \frac{0.0366}{-0.354978} \approx -0.103 \]

### Conclusion
The value of the expression is approximately \(-0.103\).