If `x=5.51,y=5.52` and `z=5.57`, then what is the value of `x^(3)+y^(3)+z^(3)-3xyz` ?

Options

A. `5.146`
B. `51.46`
C. `0.05146`
D. `0.5146`

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 - 3xyz \) given \( x = 5.51 \), \( y = 5.52 \), and \( z = 5.57 \).

### Step-by-Step Solution:

1. **Identify the Values**:
We have:
\[ x = 5.51, \quad y = 5.52, \quad z = 5.57 \]

2. **Calculate \( x + y + z \)**:
\[ x + y + z = 5.51 + 5.52 + 5.57 = 16.60 \]

3. **Calculate \( xy + yz + zx \)**:
\[ xy = 5.51 \times 5.52 = 30.49 \quad (approx) \]
\[ yz = 5.52 \times 5.57 = 30.74 \quad (approx) \]
\[ zx = 5.57 \times 5.51 = 30.68 \quad (approx) \]
Now, sum these products:
\[ xy + yz + zx \approx 30.49 + 30.74 + 30.68 = 91.91 \]

4. **Calculate \( xyz \)**:
\[ xyz = 5.51 \times 5.52 \times 5.57 \approx 169.58 \quad (approx) \]

5. **Use the Identity**:
We can use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)((x + y + z)^2 - 3(xy + yz + zx)) \]
Substitute the values we calculated:
\[ x^3 + y^3 + z^3 - 3xyz = 16.60 \left( 16.60^2 - 3 \times 91.91 \right) \]

6. **Calculate \( 16.60^2 \)**:
\[ 16.60^2 = 275.56 \]

7. **Calculate \( 3 \times 91.91 \)**:
\[ 3 \times 91.91 = 275.73 \]

8. **Final Calculation**:
\[ x^3 + y^3 + z^3 - 3xyz = 16.60 \left( 275.56 - 275.73 \right) = 16.60 \times (-0.17) \approx -2.83 \]

### Final Answer:
\[ x^3 + y^3 + z^3 - 3xyz \approx -2.83 \]