If `x = 2 + sqrt(5)` then the value of `x^(3) + x^(-3)` is :

Options

A. `40sqrt(5)`
B. `34sqrt(5)`
C. 52
D. `36sqrt(5)`

Answer

To find the value of \( x^3 + x^{-3} \) where \( x = 2 + \sqrt{5} \), we can follow these steps:

### Step 1: Calculate \( x^{-1} \)
First, we need to find \( x^{-1} \):
\[ x^{-1} = \frac{1}{x} = \frac{1}{2 + \sqrt{5}} \]
To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ x^{-1} = \frac{1 \cdot (2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} = \frac{2 - \sqrt{5}}{4 - 5} = 2 - \sqrt{5} \]

### Step 2: Calculate \( x + x^{-1} \)
Now, we can find \( x + x^{-1} \):
\[ x + x^{-1} = (2 + \sqrt{5}) + (2 - \sqrt{5}) = 4 \]

### Step 3: Use the identity for cubes
We can use the identity for cubes:
\[ x^3 + x^{-3} = (x + x^{-1})^3 - 3(x + x^{-1}) \]
Substituting \( x + x^{-1} = 4 \):
\[ x^3 + x^{-3} = 4^3 - 3 \cdot 4 \]

### Step 4: Calculate \( 4^3 \) and \( 3 \cdot 4 \)
Calculating \( 4^3 \):
\[ 4^3 = 64 \]
Calculating \( 3 \cdot 4 \):
\[ 3 \cdot 4 = 12 \]

### Step 5: Substitute back into the equation
Now substituting these values back:
\[ x^3 + x^{-3} = 64 - 12 = 52 \]

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

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