If x + 1/x = 2√3, then find the value of x³ + 1/x³.

Question

If x + 1/x = 2√3, then find the value of x³ + 1/x³.

1. A. 10√3
2. B. 6√3
3. C. 18√3 (Correct answer)
4. D. 24√3

Correct Answer

Option C — 18√3

Detailed Solution & Explanation

The correct answer is 18√3.

Key Points

• Use the standard identity rather than solving for x:
  • x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)
• Substituting x + 1/x = 2√3:
  • (2√3)³ = 8 × 3√3 = 24√3
  • 3 × 2√3 = 6√3
• So x³ + 1/x³ = 24√3 − 6√3 = 18√3.

Additional Information

• The identity family is worth memorising outright, with a = x + 1/x:
  • x² + 1/x² = a² − 2
  • x³ + 1/x³ = a³ − 3a
  • and if instead x − 1/x = b, then x² + 1/x² = b² + 2
• Here x² + 1/x² would be (2√3)² − 2 = 12 − 2 = 10, a common follow-up.
• Note (2√3)³ = 2³ × (√3)³ = 8 × 3√3; cubing the surd correctly is where this question is usually lost.
• Solving the quadratic x² − 2√3x + 1 = 0 for x and cubing directly gives the same answer but takes far longer.