If `sqrt(8)x^(6)+3 sqrt(3)y^(6)=(Ax^(2)+By^(2))(Cx^(4)+Dx^(2)y^(2)+3y^(4))`, then the value of `((C-D))/((A+B))` is

Options

A. A)`2 sqrt(2)`
B. B)`sqrt(2)`
C. C)`sqrt(3)`
D. D)`3 sqrt(3)`

Answer

To solve the equation \( \sqrt{8}x^6 + 3\sqrt{3}y^6 = (Ax^2 + By^2)(Cx^4 + Dx^2y^2 + 3y^4) \), we will start by factorizing the left-hand side and then comparing coefficients to find the values of \( A \), \( B \), \( C \), and \( D \).

### Step 1: Factor the left-hand side
The expression \( \sqrt{8}x^6 + 3\sqrt{3}y^6 \) can be recognized as a sum of cubes. We can rewrite it as:
\[ (\sqrt{2}x^2)^3 + (\sqrt{3}y^2)^3 \]
Using the sum of cubes formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \), we set \( a = \sqrt{2}x^2 \) and \( b = \sqrt{3}y^2 \).

### Step 2: Apply the sum of cubes formula
Applying the formula, we get:
\[ (\sqrt{2}x^2 + \sqrt{3}y^2)\left((\sqrt{2}x^2)^2 - (\sqrt{2}x^2)(\sqrt{3}y^2) + (\sqrt{3}y^2)^2\right) \]
Calculating each term:
- The first term: \( (\sqrt{2}x^2)^2 = 2x^4 \)
- The second term: \( -(\sqrt{2}x^2)(\sqrt{3}y^2) = -\sqrt{6}x^2y^2 \)
- The third term: \( (\sqrt{3}y^2)^2 = 3y^4 \)

Thus, we can rewrite the expression as:
\[ (\sqrt{2}x^2 + \sqrt{3}y^2)(2x^4 - \sqrt{6}x^2y^2 + 3y^4) \]

### Step 3: Compare coefficients
From the factorization, we can identify:
- \( A = \sqrt{2} \)
- \( B = \sqrt{3} \)
- \( C = 2 \)
- \( D = -\sqrt{6} \)

### Step 4: Calculate \( C - D \) and \( A + B \)
Now we need to calculate \( C - D \) and \( A + B \):
\[ C - D = 2 - (-\sqrt{6}) = 2 + \sqrt{6} \]
\[ A + B = \sqrt{2} + \sqrt{3} \]

### Step 5: Find \( \frac{C - D}{A + B} \)
Now we can find:
\[ \frac{C - D}{A + B} = \frac{2 + \sqrt{6}}{\sqrt{2} + \sqrt{3}} \]

### Step 6: Simplify the expression
To simplify, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(2 + \sqrt{6})(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} \]
Calculating the denominator:
\[ (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 \]
Calculating the numerator:
\[ (2 + \sqrt{6})(\sqrt{2} - \sqrt{3}) = 2\sqrt{2} - 2\sqrt{3} + \sqrt{12} - \sqrt{18} = 2\sqrt{2} - 2\sqrt{3} + 2\sqrt{3} - 3\sqrt{2} = -\sqrt{2} \]
Thus, we have:
\[ \frac{-\sqrt{2}}{-1} = \sqrt{2} \]

### Final Answer
The value of \( \frac{C - D}{A + B} \) is \( \sqrt{2} \).