If `x =2, y =1 and z=-3,` then `x ^(3) + y ^(3) + z ^(3)- 3xyz` is equal to :

Options

A. 0
B. 2
C. 6
D. 8

Answer

To solve the expression \( x^3 + y^3 + z^3 - 3xyz \) given \( x = 2 \), \( y = 1 \), and \( z = -3 \), we will follow these steps:

### Step 1: Substitute the values of \( x \), \( y \), and \( z \)
We start by substituting the values into the expression:
\[ x^3 + y^3 + z^3 - 3xyz = 2^3 + 1^3 + (-3)^3 - 3 \cdot 2 \cdot 1 \cdot (-3) \]

### Step 2: Calculate each term
Now we calculate each term:
- \( 2^3 = 8 \)
- \( 1^3 = 1 \)
- \( (-3)^3 = -27 \)
- \( 3 \cdot 2 \cdot 1 \cdot (-3) = -18 \)

### Step 3: Substitute the calculated values back into the expression
Now we substitute these calculated values back into the expression:
\[ 8 + 1 - 27 - (-18) \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ 8 + 1 - 27 + 18 \]

### Step 5: Perform the addition and subtraction
Now we perform the addition and subtraction step by step:
1. \( 8 + 1 = 9 \)
2. \( 9 - 27 = -18 \)
3. \( -18 + 18 = 0 \)

### Final Result
Thus, the value of the expression \( x^3 + y^3 + z^3 - 3xyz \) is:
\[ \boxed{0} \]