If `x =3 + sqrt2:y = 3 - sqrt2,` then the value of `(x ^(2) + y ^(2))/( x ^(3) + y ^(3)): `

Options

A. `(11)/(45)`
B. `(5)/(9)`
C. `(67)/(157)`
D. `(61)/(540)`

Answer

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

### Step 1: Calculate \(x^2\)
\[ x^2 = (3 + \sqrt{2})^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{2} + (\sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2} \]

### Step 2: Calculate \(y^2\)
\[ y^2 = (3 - \sqrt{2})^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{2} + (\sqrt{2})^2 = 9 - 6\sqrt{2} + 2 = 11 - 6\sqrt{2} \]

### Step 3: Calculate \(x^2 + y^2\)
\[ x^2 + y^2 = (11 + 6\sqrt{2}) + (11 - 6\sqrt{2}) = 22 \]

### Step 4: Calculate \(x^3\)
Using the binomial expansion:
\[ x^3 = (3 + \sqrt{2})^3 = 3^3 + 3 \cdot 3^2 \cdot \sqrt{2} + 3 \cdot 3 \cdot (\sqrt{2})^2 + (\sqrt{2})^3 \]
Calculating each term:
\[ = 27 + 27\sqrt{2} + 18 + 2\sqrt{2} = 29 + 29\sqrt{2} \]

### Step 5: Calculate \(y^3\)
Using the binomial expansion:
\[ y^3 = (3 - \sqrt{2})^3 = 3^3 - 3 \cdot 3^2 \cdot \sqrt{2} + 3 \cdot 3 \cdot (\sqrt{2})^2 - (\sqrt{2})^3 \]
Calculating each term:
\[ = 27 - 27\sqrt{2} + 18 - 2\sqrt{2} = 45 - 29\sqrt{2} \]

### Step 6: Calculate \(x^3 + y^3\)
\[ x^3 + y^3 = (29 + 29\sqrt{2}) + (45 - 29\sqrt{2}) = 74 \]

### Step 7: Calculate \(\frac{x^2 + y^2}{x^3 + y^3}\)
\[ \frac{x^2 + y^2}{x^3 + y^3} = \frac{22}{74} = \frac{11}{37} \]

Thus, the final answer is:
\[ \frac{x^2 + y^2}{x^3 + y^3} = \frac{11}{37} \]