If `p + q = 6 and pq = 8 ` , then `p^(3) + q^(3)` is equal to

Options

A. 216
B. 144
C. 72
D. 36

Answer

To solve the problem, we need to find the value of \( p^3 + q^3 \) given that \( p + q = 6 \) and \( pq = 8 \).

### Step-by-step Solution:

1. **Use the identity for \( p^3 + q^3 \)**:
The identity states that:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
We need to find \( p^2 - pq + q^2 \).

2. **Calculate \( p^2 + q^2 \)**:
We can find \( p^2 + q^2 \) using the square of the sum:
\[ (p + q)^2 = p^2 + q^2 + 2pq \]
Substituting the known values:
\[ 6^2 = p^2 + q^2 + 2 \cdot 8 \]
This simplifies to:
\[ 36 = p^2 + q^2 + 16 \]
Therefore:
\[ p^2 + q^2 = 36 - 16 = 20 \]

3. **Substitute into the identity**:
Now we can substitute \( p^2 + q^2 \) into the expression for \( p^2 - pq + q^2 \):
\[ p^2 - pq + q^2 = (p^2 + q^2) - pq = 20 - 8 = 12 \]

4. **Calculate \( p^3 + q^3 \)**:
Now we substitute back into the identity:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) = 6 \cdot 12 = 72 \]

Thus, the value of \( p^3 + q^3 \) is \( \boxed{72} \).