Explanation

(x⁴ + 1/x⁴)² = x⁸ + 2 + 1/x⁸ = 49 gives x⁴ + 1/x⁴ = 7. Then (x² + 1/x²)² = 7 + 2 = 9 gives x² + 1/x² = 3. Finally x⁶ + 1/x⁶ = (x² + 1/x²)³ − 3(x² + 1/x²) = 27 − 9 = 18.