If `x = sqrt3 + sqrt2 and y = sqrt3 - sqrt2 ,` then the value of `8 xy (x ^(2) + y ^(2))` is :

Options

A. `16 sqrt6`
B. `32`
C. `48 sqrt2`
D. `80`

Answer

To solve the problem, we need to find the value of \( 8xy(x^2 + y^2) \) given \( x = \sqrt{3} + \sqrt{2} \) and \( y = \sqrt{3} - \sqrt{2} \).

### Step 1: Calculate \( x^2 \) and \( y^2 \)

1. **Calculate \( x^2 \)**:
\[ x^2 = (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{3}\sqrt{2} = 5 + 2\sqrt{6} \]

2. **Calculate \( y^2 \)**:
\[ y^2 = (\sqrt{3} - \sqrt{2})^2 = 3 + 2 - 2\sqrt{3}\sqrt{2} = 5 - 2\sqrt{6} \]

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

Now, we can add \( x^2 \) and \( y^2 \):
\[ x^2 + y^2 = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10 \]

### Step 3: Calculate \( xy \)

Next, we calculate \( xy \):
\[ xy = (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

### Step 4: Substitute into the expression

Now, we can substitute \( xy \) and \( x^2 + y^2 \) into the expression \( 8xy(x^2 + y^2) \):
\[ 8xy(x^2 + y^2) = 8 \cdot 1 \cdot 10 = 80 \]

### Final Answer

Thus, the value of \( 8xy(x^2 + y^2) \) is \( \boxed{80} \).

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