If x + 1/x = 6 (x ≠ 0), evaluate x^4 + 1/x^4 efficiently using power-sum identities.

Introduction / Context:Many olympiad-style and aptitude questions rely on the chain of identities connecting x + 1/x to higher-power symmetric sums. Mastering these shortcuts saves time and prevents arithmetic mishaps.

Given Data / Assumptions:

• x + 1/x = 6, x ≠ 0.
• Goal: compute x^4 + 1/x^4.

Concept / Approach:First find x^2 + 1/x^2 from (x + 1/x)^2 − 2. Then use the identity (x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2 to reach the target without expanding powers directly.

Step-by-Step Solution:x^2 + 1/x^2 = (x + 1/x)^2 − 2 = 6^2 − 2 = 36 − 2 = 34.Then x^4 + 1/x^4 = (x^2 + 1/x^2)^2 − 2 = 34^2 − 2 = 1156 − 2 = 1154.

Verification / Alternative check:Using the recurrence S_n = (x + 1/x)S_{n−1} − S_{n−2}, with S_1 = 6 and S_2 = 34, gives S_4 = 6*(6*34 − 6) − 34 = 6*(204 − 6) − 34 = 6*198 − 34 = 1188 − 34 = 1154, confirming the result.

Why Other Options Are Wrong:

• 1152, 1150, 1148, 1138: Each is close, typically caused by forgetting to subtract 2 in the final step or squaring errors.

Common Pitfalls:Expanding x^4 directly; dropping the constant 2 when applying (x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2.

Final Answer:1154