If `x=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))` and `y=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` find the value of `x^(3)+y^(3)`

Options

A. `807`
B. `907`
C. `970`
D. `870`

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 \) given:

\[ x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \]
\[ y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \]

### Step 1: Find \( x + y \)

To find \( x + y \), we add the two fractions:

\[ x + y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \]

To add these fractions, we need a common denominator, which is \( (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) \).

### Step 2: Calculate the common denominator

The common denominator simplifies as follows:

\[ (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1 \]

### Step 3: Combine the fractions

Now we can combine the fractions:

\[ x + y = \frac{(\sqrt{3} - \sqrt{2})^2 + (\sqrt{3} + \sqrt{2})^2}{1} \]

### Step 4: Expand the numerators

Now we expand both squares:

\[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \]
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]

### Step 5: Add the expanded forms

Adding these two results:

\[ x + y = (5 - 2\sqrt{6}) + (5 + 2\sqrt{6}) = 10 \]

### Step 6: Find \( xy \)

Next, we find \( xy \):

\[ xy = \left(\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\right) \left(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\right) = 1 \]

### Step 7: Use the identity for \( x^3 + y^3 \)

We can use the identity:

\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]

We already have \( x + y = 10 \) and \( xy = 1 \). Now we need \( x^2 + y^2 \):

\[ x^2 + y^2 = (x + y)^2 - 2xy = 10^2 - 2 \cdot 1 = 100 - 2 = 98 \]

### Step 8: Substitute into the identity

Now we substitute into the identity:

\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 10(98 - 1) = 10 \cdot 97 = 970 \]

### Final Answer

Thus, the value of \( x^3 + y^3 \) is:

\[ \boxed{970} \]