Simplify :
`((1.5)^3 + (4.7)^3+(3.8)^3 -3xx 1.5 xx 4.7 xx 3.8)/((1.5)^2 +(4.7)^2+(3.8)^2-1.5 xx 4.7 -4.7 xx 3.8 -3.8 xx 1.5)`

Options

A. 0
B. 1
C. 10
D. 30

Answer

To simplify the expression

\[ \frac{(1.5)^3 + (4.7)^3 + (3.8)^3 - 3 \cdot 1.5 \cdot 4.7 \cdot 3.8}{(1.5)^2 + (4.7)^2 + (3.8)^2 - 1.5 \cdot 4.7 - 4.7 \cdot 3.8 - 3.8 \cdot 1.5} \]

we can use the identity for the sum of cubes and the formula for the difference of squares.

### Step 1: Recognize the Identity
The numerator can be recognized as the formula for the sum of cubes:
\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA) \]
where \( A = 1.5 \), \( B = 4.7 \), and \( C = 3.8 \).

### Step 2: Calculate \( A + B + C \)
Now, we calculate:
\[ A + B + C = 1.5 + 4.7 + 3.8 \]
Calculating this gives:
\[ 1.5 + 4.7 = 6.2 \]
\[ 6.2 + 3.8 = 10 \]

### Step 3: Calculate \( A^2 + B^2 + C^2 - AB - BC - CA \)
Next, we compute the denominator:
\[ A^2 + B^2 + C^2 - AB - BC - CA \]
Calculating \( A^2 \), \( B^2 \), and \( C^2 \):
\[ (1.5)^2 = 2.25, \quad (4.7)^2 = 22.09, \quad (3.8)^2 = 14.44 \]
Adding these:
\[ 2.25 + 22.09 + 14.44 = 38.78 \]

Now, calculate \( AB \), \( BC \), and \( CA \):
\[ AB = 1.5 \cdot 4.7 = 7.05, \quad BC = 4.7 \cdot 3.8 = 17.86, \quad CA = 3.8 \cdot 1.5 = 5.7 \]
Adding these:
\[ 7.05 + 17.86 + 5.7 = 30.61 \]

Now substitute back into the expression:
\[ A^2 + B^2 + C^2 - AB - BC - CA = 38.78 - 30.61 = 8.17 \]

### Step 4: Substitute Back into the Original Expression
Now, we substitute back into the original expression:
\[ \frac{(A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA)}{A^2 + B^2 + C^2 - AB - BC - CA} \]
This simplifies to:
\[ \frac{10 \cdot 8.17}{8.17} = 10 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{10} \]