Explanation:

Let’s break it down step by step:

- You’re given x + (1/x) = 3v2.

- The key is to build up higher powers recursively.

- Start with the basic formula:

(x + 1/x) = a

(x² + 1/x²) = a² – 2

(x³ + 1/x³) = a³ – 3a

and so on.

- For a = 3v2:

x + 1/x = 3v2

x² + 1/x² = (3v2)² – 2 = 18 – 2 = 16

x³ + 1/x³ = (3v2)³ – 3*(3v2) = 54v2 – 9v2 = 45v2

x4 + 1/x4 = (x² + 1/x²)² – 2 = 16² – 2 = 254

x5 + 1/x5 = (x4 + 1/x4)*(x + 1/x) – (x³ + 1/x³)

= 254*(3v2) – 45v2

= (254×3 – 45)v2

= (762 – 45)v2

= 717v2

- Putting it together:

- Option 1: 178v3 (not our answer)

- Option 2: 789v2 (not our answer)

- Option 3: 1581v2 (not our answer)

- Option 4: 717v2 (this one matches perfectly!)