If `x + y =3 sqrt8, x - y = sqrt2,` then the value of 8 xy `(x ^(2) + y ^(2))` is

Options

A. 5760
B. 5440
C. 5360
D. 5180

Answer

To solve the problem, we need to find the value of \( 8xy(x^2 + y^2) \) given the equations:

1. \( x + y = 3\sqrt{8} \)
2. \( x - y = \sqrt{2} \)

### Step 1: Solve for \( x \) and \( y \)

We can add and subtract the two equations to find \( x \) and \( y \).

Adding the two equations:
\[ (x + y) + (x - y) = 3\sqrt{8} + \sqrt{2} \]
This simplifies to:
\[ 2x = 3\sqrt{8} + \sqrt{2} \]
Thus,
\[ x = \frac{3\sqrt{8} + \sqrt{2}}{2} \]

Now, subtracting the second equation from the first:
\[ (x + y) - (x - y) = 3\sqrt{8} - \sqrt{2} \]
This simplifies to:
\[ 2y = 3\sqrt{8} - \sqrt{2} \]
Thus,
\[ y = \frac{3\sqrt{8} - \sqrt{2}}{2} \]

### Step 2: Calculate \( xy \)

Using the identity \( (x+y)(x-y) = x^2 - y^2 \):
\[ xy = \frac{(x+y)(x-y)}{2} \]
Substituting the values:
\[ xy = \frac{(3\sqrt{8})(\sqrt{2})}{2} \]
Calculating this gives:
\[ xy = \frac{3\sqrt{16}}{2} = \frac{3 \cdot 4}{2} = 6 \]

### Step 3: Calculate \( x^2 + y^2 \)

Using the identity \( x^2 + y^2 = (x+y)^2 - 2xy \):
\[ x^2 + y^2 = (3\sqrt{8})^2 - 2(6) \]
Calculating this gives:
\[ x^2 + y^2 = 72 - 12 = 60 \]

### Step 4: Calculate \( 8xy(x^2 + y^2) \)

Now we can find \( 8xy(x^2 + y^2) \):
\[ 8xy(x^2 + y^2) = 8 \cdot 6 \cdot 60 \]
Calculating this gives:
\[ 8 \cdot 6 = 48 \]
\[ 48 \cdot 60 = 2880 \]

Thus, the final answer is:
\[ \boxed{2880} \]