If `x^3 + y^3 = 405` and `x+ y = 9` , then find the value of xy.

Options

A. 12
B. 8
C. 18
D. 24

Answer

To solve the problem, we need to find the value of \( xy \) given the equations \( x^3 + y^3 = 405 \) and \( x + y = 9 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
We know that:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We can substitute \( x + y = 9 \) into this identity.

2. **Substitute the known value**:
Substitute \( x + y = 9 \) into the equation:
\[ x^3 + y^3 = 9(x^2 - xy + y^2) \]
Given \( x^3 + y^3 = 405 \), we can set up the equation:
\[ 405 = 9(x^2 - xy + y^2) \]

3. **Simplify the equation**:
Divide both sides by 9:
\[ x^2 - xy + y^2 = \frac{405}{9} = 45 \]

4. **Express \( x^2 + y^2 \) in terms of \( xy \)**:
We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting \( x + y = 9 \):
\[ x^2 + y^2 = 9^2 - 2xy = 81 - 2xy \]

5. **Substitute into the equation**:
Now substitute \( x^2 + y^2 \) into the equation from step 3:
\[ (81 - 2xy) - xy = 45 \]
Simplifying gives:
\[ 81 - 3xy = 45 \]

6. **Solve for \( xy \)**:
Rearranging the equation:
\[ 3xy = 81 - 45 \]
\[ 3xy = 36 \]
Dividing both sides by 3:
\[ xy = \frac{36}{3} = 12 \]

### Final Answer:
The value of \( xy \) is \( 12 \).