If `p + m = 6 and p ^(3) + m ^(3) = 72,` then the value of pm is :

Options

A. 12
B. 8
C. 6
D. 9

Answer

To solve the problem, we need to find the value of \( pm \) given the equations \( p + m = 6 \) and \( p^3 + m^3 = 72 \).

### Step-by-step solution:

1. **Use the identity for the sum of cubes**:
The identity for the sum of cubes states that:
\[ p^3 + m^3 = (p + m)(p^2 - pm + m^2) \]
We know \( p + m = 6 \).

2. **Substitute \( p + m \) into the identity**:
\[ p^3 + m^3 = 6(p^2 - pm + m^2) \]

3. **Express \( p^2 + m^2 \)**:
We can express \( p^2 + m^2 \) using the square of the sum:
\[ p^2 + m^2 = (p + m)^2 - 2pm = 6^2 - 2pm = 36 - 2pm \]

4. **Substitute \( p^2 + m^2 \) back into the equation**:
Now we substitute \( p^2 + m^2 \) into the identity:
\[ p^3 + m^3 = 6((36 - 2pm) - pm) = 6(36 - 3pm) \]

5. **Set the equation equal to \( p^3 + m^3 = 72 \)**:
\[ 6(36 - 3pm) = 72 \]

6. **Divide both sides by 6**:
\[ 36 - 3pm = 12 \]

7. **Rearrange the equation**:
\[ 36 - 12 = 3pm \implies 24 = 3pm \]

8. **Solve for \( pm \)**:
\[ pm = \frac{24}{3} = 8 \]

Thus, the value of \( pm \) is \( 8 \).