If `x = 2 +sqrt3, y = 2 - sqrt3` then the value `(x^2+y^2)/(x^3+y^3)` is

Options

A. `7/38`
B. `7/40`
C. `7/18
D. `7/26`

Answer

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

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

First, we find \(x + y\) and \(xy\):

\[ x + y = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]

\[ xy = (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

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

Using the identity \(x^2 + y^2 = (x + y)^2 - 2xy\):

\[ x^2 + y^2 = (4)^2 - 2(1) = 16 - 2 = 14 \]

### Step 3: Calculate \(x^3 + y^3\)

Using the identity \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\):

First, we need \(x^2 - xy + y^2\):

\[ x^2 - xy + y^2 = (x^2 + y^2) - xy = 14 - 1 = 13 \]

Now we can calculate \(x^3 + y^3\):

\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 4 \cdot 13 = 52 \]

### Step 4: Calculate \(\frac{x^2 + y^2}{x^3 + y^3}\)

Now we can find the value of \(\frac{x^2 + y^2}{x^3 + y^3}\):

\[ \frac{x^2 + y^2}{x^3 + y^3} = \frac{14}{52} = \frac{7}{26} \]

### Final Answer

Thus, the value of \(\frac{x^2 + y^2}{x^3 + y^3}\) is \(\frac{7}{26}\).

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