If `x^(2) - sqrt(7) x +1 =0` then `(x^(3)+x^(-3))` =?

Options

A. `10 sqrt(7)`
B. `4 sqrt(7)`
C. `7 sqrt(7)`
D. `3 sqrt(7)`

Answer

To solve the equation \( x^2 - \sqrt{7}x + 1 = 0 \) and find the value of \( x^3 + x^{-3} \), we can follow these steps:

### Step 1: Solve for \( x + \frac{1}{x} \)

Starting from the given equation:
\[ x^2 - \sqrt{7}x + 1 = 0 \]
We can rearrange it to isolate the \( \sqrt{7}x \) term:
\[ x^2 + 1 = \sqrt{7}x \]
Next, divide the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ x + \frac{1}{x} = \sqrt{7} \]

### Step 2: Use the identity for \( x^3 + x^{-3} \)

We know the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3\left( x + \frac{1}{x} \right) \]
Substituting \( x + \frac{1}{x} = \sqrt{7} \) into the identity:
\[ x^3 + \frac{1}{x^3} = \left( \sqrt{7} \right)^3 - 3\sqrt{7} \]

### Step 3: Calculate \( \left( \sqrt{7} \right)^3 \)

Calculating \( \left( \sqrt{7} \right)^3 \):
\[ \left( \sqrt{7} \right)^3 = 7\sqrt{7} \]

### Step 4: Substitute back into the equation

Now substitute this back into the equation:
\[ x^3 + \frac{1}{x^3} = 7\sqrt{7} - 3\sqrt{7} \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} = (7 - 3)\sqrt{7} = 4\sqrt{7} \]

### Final Answer

Thus, the value of \( x^3 + x^{-3} \) is:
\[ \boxed{4\sqrt{7}} \]