If `x+y+z=17,xy+yz+zx=85` , then what is the value of `sqrt(x^(3)+y^(3)+z^(3)-3xyz)`?

Options

A. `11sqrt(3)`
B. `17sqrt(2)`
C. `17sqrt(3)`
D. `11sqrt(2)`

Answer

To solve the problem, we need to find the value of \( \sqrt{x^3 + y^3 + z^3 - 3xyz} \) given the equations \( x + y + z = 17 \) and \( xy + yz + zx = 85 \).

### Step 1: Use the identity for \( x^3 + y^3 + z^3 - 3xyz \)

We can use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x+y+z) \left( (x+y+z)^2 - 3(xy + yz + zx) \right) \]

### Step 2: Substitute the known values

From the problem, we know:
- \( x + y + z = 17 \)
- \( xy + yz + zx = 85 \)

Now, substitute these values into the identity:
\[ x^3 + y^3 + z^3 - 3xyz = 17 \left( 17^2 - 3 \cdot 85 \right) \]

### Step 3: Calculate \( 17^2 \) and \( 3 \cdot 85 \)

First, calculate \( 17^2 \):
\[ 17^2 = 289 \]

Next, calculate \( 3 \cdot 85 \):
\[ 3 \cdot 85 = 255 \]

### Step 4: Substitute these results back into the equation

Now substitute these values into the equation:
\[ x^3 + y^3 + z^3 - 3xyz = 17 \left( 289 - 255 \right) \]

### Step 5: Simplify the expression inside the parentheses

Now simplify:
\[ 289 - 255 = 34 \]

So we have:
\[ x^3 + y^3 + z^3 - 3xyz = 17 \cdot 34 \]

### Step 6: Calculate \( 17 \cdot 34 \)

Now calculate:
\[ 17 \cdot 34 = 578 \]

### Step 7: Find the square root

Finally, we need to find:
\[ \sqrt{x^3 + y^3 + z^3 - 3xyz} = \sqrt{578} \]

### Step 8: Simplify \( \sqrt{578} \)

We can simplify \( \sqrt{578} \):
\[ \sqrt{578} = \sqrt{2 \cdot 289} = \sqrt{2} \cdot \sqrt{289} = \sqrt{2} \cdot 17 = 17\sqrt{2} \]

Thus, the final answer is:
\[ \sqrt{x^3 + y^3 + z^3 - 3xyz} = 17\sqrt{2} \]