Simplify : `((1.5)^(3)+(4.7)^(3)+(3.8)^(3)-3xx1.5xx4.7xx3.8)/((1.5)^(2)+(4.7)^(2)+(3.8)^(2)-1.5xx4.7-4.7xx3.8-3.8xx1.5)`

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 identities related to the sum of cubes and the sum of squares.

### Step 1: Identify Variables
Let:
- \( A = 1.5 \)
- \( B = 4.7 \)
- \( C = 3.8 \)

### Step 2: Apply the Sum of Cubes Formula
The numerator can be expressed using the identity for the sum of cubes:

\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA) \]

So, we rewrite the numerator as:

\[ (A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA) \]

### Step 3: Calculate \( A + B + C \)
Now, calculate \( A + B + C \):

\[ A + B + C = 1.5 + 4.7 + 3.8 = 10 \]

### Step 4: Calculate \( A^2 + B^2 + C^2 \)
Next, we calculate \( A^2 + B^2 + C^2 \):

\[ A^2 = (1.5)^2 = 2.25 \]
\[ B^2 = (4.7)^2 = 22.09 \]
\[ C^2 = (3.8)^2 = 14.44 \]

Thus,

\[ A^2 + B^2 + C^2 = 2.25 + 22.09 + 14.44 = 38.78 \]

### Step 5: Calculate \( AB + BC + CA \)
Now, calculate \( AB + BC + CA \):

\[ AB = 1.5 \cdot 4.7 = 7.05 \]
\[ BC = 4.7 \cdot 3.8 = 17.86 \]
\[ CA = 3.8 \cdot 1.5 = 5.7 \]

Thus,

\[ AB + BC + CA = 7.05 + 17.86 + 5.7 = 30.61 \]

### Step 6: Calculate \( A^2 + B^2 + C^2 - AB - BC - CA \)
Now substitute into the expression:

\[ A^2 + B^2 + C^2 - AB - BC - CA = 38.78 - 30.61 = 8.17 \]

### Step 7: Substitute Back into the Expression
Now, substitute back into the numerator:

\[ \text{Numerator} = (A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA) = 10 \cdot 8.17 = 81.7 \]

### Step 8: Calculate the Denominator
Now, we calculate the denominator:

\[ A^2 + B^2 + C^2 - AB - BC - CA = 38.78 - 30.61 = 8.17 \]

### Step 9: Final Simplification
Now we can simplify the entire expression:

\[ \frac{81.7}{8.17} = 10 \]

Thus, the simplified value of the expression is:

\[ \boxed{10} \]