If `x= 3 sqrt( 28) , y = 3 sqrt(27)` then the value of ` x - y - ( 1)/( x^(2) + xy +y^(2)) ` is

Options

A. 0
B. 7
C. 6
D. 5

Answer

To solve the problem, we need to find the value of the expression:

\[ x - y - \frac{1}{x^2 + xy + y^2} \]

where \( x = 3 \sqrt{28} \) and \( y = 3 \sqrt{27} \).

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

First, we will calculate the values of \( x \) and \( y \).

\[ x = 3 \sqrt{28} = 3 \cdot \sqrt{4 \cdot 7} = 3 \cdot 2 \sqrt{7} = 6 \sqrt{7} \]

\[ y = 3 \sqrt{27} = 3 \cdot \sqrt{9 \cdot 3} = 3 \cdot 3 \sqrt{3} = 9 \sqrt{3} \]

### Step 2: Calculate \( x - y \)

Now we find \( x - y \):

\[ x - y = 6 \sqrt{7} - 9 \sqrt{3} \]

### Step 3: Calculate \( x^2, y^2, \) and \( xy \)

Next, we calculate \( x^2 \), \( y^2 \), and \( xy \):

\[ x^2 = (6 \sqrt{7})^2 = 36 \cdot 7 = 252 \]

\[ y^2 = (9 \sqrt{3})^2 = 81 \cdot 3 = 243 \]

\[ xy = (6 \sqrt{7})(9 \sqrt{3}) = 54 \sqrt{21} \]

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

Now we can find \( x^2 + xy + y^2 \):

\[ x^2 + y^2 = 252 + 243 = 495 \]

Thus,

\[ x^2 + xy + y^2 = 495 + 54 \sqrt{21} \]

### Step 5: Calculate \( \frac{1}{x^2 + xy + y^2} \)

Now we need to find \( \frac{1}{x^2 + xy + y^2} \):

\[ \frac{1}{x^2 + xy + y^2} = \frac{1}{495 + 54 \sqrt{21}} \]

### Step 6: Combine all parts to find the final expression

Now we substitute back into our expression:

\[ x - y - \frac{1}{x^2 + xy + y^2} = (6 \sqrt{7} - 9 \sqrt{3}) - \frac{1}{495 + 54 \sqrt{21}} \]

This is the final expression we need to evaluate.

### Final Result

The final value of the expression \( x - y - \frac{1}{x^2 + xy + y^2} \) is:

\[ 6 \sqrt{7} - 9 \sqrt{3} - \frac{1}{495 + 54 \sqrt{21}} \]