If `x+y+z=5 and xy+yz+zx=7`, then `x^(3)+y^(3)+z^(3)-3xyz` = ___________.

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 - 3xyz \) given the equations \( x + y + z = 5 \) and \( xy + yz + zx = 7 \).

### Step-by-Step Solution:

1. **Use the Identity**:
We can use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z) \left( x^2 + y^2 + z^2 - xy - yz - zx \right) \]
We will denote this as Equation (1).

2. **Substituting Known Values**:
From the problem, we know:
\[ x + y + z = 5 \]
We need to find \( x^2 + y^2 + z^2 \). We can find this using the square of the sum:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]
Substituting the known values:
\[ 5^2 = x^2 + y^2 + z^2 + 2 \cdot 7 \]
This simplifies to:
\[ 25 = x^2 + y^2 + z^2 + 14 \]

3. **Solving for \( x^2 + y^2 + z^2 \)**:
Rearranging the equation gives:
\[ x^2 + y^2 + z^2 = 25 - 14 = 11 \]

4. **Substituting Back into the Identity**:
Now we substitute \( x^2 + y^2 + z^2 \) back into Equation (1):
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z) \left( x^2 + y^2 + z^2 - (xy + yz + zx) \right) \]
Substituting the values we have:
\[ x^3 + y^3 + z^3 - 3xyz = 5 \left( 11 - 7 \right) \]

5. **Calculating the Final Value**:
This simplifies to:
\[ x^3 + y^3 + z^3 - 3xyz = 5 \cdot 4 = 20 \]

Thus, the final answer is:
\[ \boxed{20} \]