Question

If \( \left( x + \frac{1}{x} \right) = \sqrt{3} \), then the value of \( \left( x^3 + \frac{1}{x^3} \right) \) will be:

• A. \( 3(\sqrt{3} + 1) \)
• B. \( \sqrt{3} \)
• C. 0
• D. \( 3(\sqrt{3} - 1) \)

Hint:

Use the identity \( \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right) \) to find expressions involving cubes.

Answer 1

The Correct Option is A

We are given that: \[ x + \frac{1}{x} = \sqrt{3} \] To find \( x^3 + \frac{1}{x^3} \), we use the identity: \[ \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right) \] Substitute \( x + \frac{1}{x} = \sqrt{3} \) into the identity: \[ \left( \sqrt{3} \right)^3 = x^3 + \frac{1}{x^3} + 3 \times \sqrt{3} \] \[ 3\sqrt{3} = x^3 + \frac{1}{x^3} + 3\sqrt{3} \] Solving for \( x^3 + \frac{1}{x^3} \): \[ x^3 + \frac{1}{x^3} = 3(\sqrt{3} + 1) \] Thus, the correct answer is \( 3(\sqrt{3} + 1) \).