The value of
`((1.5)^3 + (4.7)^3 + (3.8)^3 - 3 xx 1.5 xx 4.7 xx 3.8)/((1.5)^2 + (4.7)^2 + (3.8)^2 - (1.5 xx 4.7) - (4.7xx 3.8) - (1.5 xx 3.8))` is :

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the given expression:

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

we can use the identity for the sum of cubes:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c) \left( a^2 + b^2 + c^2 - ab - ac - bc \right) \]

### Step 1: Identify \(a\), \(b\), and \(c\)

Let:
- \(a = 1.5\)
- \(b = 4.7\)
- \(c = 3.8\)

### Step 2: Calculate \(a + b + c\)

\[ a + b + c = 1.5 + 4.7 + 3.8 \]

Calculating this gives:

\[ 1.5 + 4.7 = 6.2 \]
\[ 6.2 + 3.8 = 10 \]

### Step 3: Calculate \(a^2 + b^2 + c^2\)

\[ a^2 + b^2 + c^2 = (1.5)^2 + (4.7)^2 + (3.8)^2 \]

Calculating each term:

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

Adding these results:

\[ 2.25 + 22.09 + 14.44 = 38.78 \]

### Step 4: Calculate \(ab + ac + bc\)

\[ ab + ac + bc = (1.5 \times 4.7) + (1.5 \times 3.8) + (4.7 \times 3.8) \]

Calculating each product:

\[ 1.5 \times 4.7 = 7.05 \]
\[ 1.5 \times 3.8 = 5.7 \]
\[ 4.7 \times 3.8 = 17.86 \]

Adding these results:

\[ 7.05 + 5.7 + 17.86 = 30.61 \]

### Step 5: Substitute into the denominator

Now substitute into the denominator:

\[ (1.5)^2 + (4.7)^2 + (3.8)^2 - (1.5 \times 4.7) - (1.5 \times 3.8) - (4.7 \times 3.8) \]

This becomes:

\[ 38.78 - 30.61 = 8.17 \]

### Step 6: Substitute into the formula

Now substitute back into the original expression:

\[ \frac{(1.5)^3 + (4.7)^3 + (3.8)^3 - 3 \times 1.5 \times 4.7 \times 3.8}{8.17} \]

Using the identity:

The numerator simplifies to:

\[ (1.5 + 4.7 + 3.8)(38.78 - 30.61) = 10 \times 8.17 = 81.7 \]

### Step 7: Final calculation

Now we can find the final value:

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

### Conclusion

Thus, the value of the expression is:

\[ \boxed{10} \]