If `x=2+ sqrt5` then the value of `x^3+x^-3` is:
यदि `x=2+sqrt 5` है तो `x^3+x^-3` का मान है :

Options

A. `40 sqrt5`
B. `34 sqrt5`
C. `46 sqrt5`
D. `36 sqrt5`

Answer

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

### Step 1: Find \( x^{-1} \)
We start by calculating \( x^{-1} \):
\[ x^{-1} = \frac{1}{x} = \frac{1}{2 + \sqrt{5}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate \( 2 - \sqrt{5} \):
\[ x^{-1} = \frac{2 - \sqrt{5}}{(2 + \sqrt{5})(2 - \sqrt{5})} \]
Calculating the denominator:
\[ (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \]
Thus,
\[ x^{-1} = -(2 - \sqrt{5}) = \sqrt{5} - 2 \]

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

### Step 3: Use the formula for \( x^3 + x^{-3} \)
We use the identity:
\[ x^3 + x^{-3} = (x + x^{-1})^3 - 3(x + x^{-1}) \]
Substituting \( k = x + x^{-1} = 2\sqrt{5} \):
\[ x^3 + x^{-3} = (2\sqrt{5})^3 - 3(2\sqrt{5}) \]
Calculating \( (2\sqrt{5})^3 \):
\[ (2\sqrt{5})^3 = 8 \cdot 5\sqrt{5} = 40\sqrt{5} \]
Now substituting back:
\[ x^3 + x^{-3} = 40\sqrt{5} - 6\sqrt{5} = 34\sqrt{5} \]

### Final Answer
Thus, the value of \( x^3 + x^{-3} \) is:
\[ \boxed{34\sqrt{5}} \]

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