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

Options

A. 42
B. 48
C. 50
D. 52

Answer

To find the value of \( x^{-3} + y^{-3} \) where \( x = 2 + \sqrt{3} \) and \( y = 2 - \sqrt{3} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \( x^{-3} + y^{-3} \) in terms of \( x \) and \( y \):
\[ x^{-3} + y^{-3} = \frac{1}{x^3} + \frac{1}{y^3} = \frac{y^3 + x^3}{x^3 y^3} \]

### Step 2: Find \( x + y \) and \( xy \)
Next, we calculate \( x + y \) 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 3: Calculate \( x^3 + y^3 \)
We use the identity for the sum of cubes:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We need \( x^2 + y^2 \), which can be found using:
\[ x^2 + y^2 = (x + y)^2 - 2xy = 4^2 - 2 \cdot 1 = 16 - 2 = 14 \]
Now we can substitute back:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 4(14 - 1) = 4 \cdot 13 = 52 \]

### Step 4: Calculate \( x^3 y^3 \)
Since \( xy = 1 \):
\[ x^3 y^3 = (xy)^3 = 1^3 = 1 \]

### Step 5: Substitute back into the expression
Now we substitute \( x^3 + y^3 \) and \( x^3 y^3 \) back into our expression:
\[ x^{-3} + y^{-3} = \frac{x^3 + y^3}{x^3 y^3} = \frac{52}{1} = 52 \]

### Final Answer
Thus, the value of \( x^{-3} + y^{-3} \) is \( \boxed{52} \).

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