Alternative approach:
a^2 + 1/a^2 = 14. (1)
From (1): a^2 = 14 - 1/a^2. Multiply through by a to get: a^3 = 14a - 1/a. (2)
From (1): 1/a^2 = 14 - a^2. Divide through by a to get: 1/a^3 = 14/a - a. (3)
Add (2) and (3): a^3 + 1/a^3 = 13(a + 1/a). (4)
Now (a + 1/a)^2 = a^2 + 1/a^2 + 2 → 14 + 2 → 16 using (1)
Hence a + 1/a = 4. (5)
Put (5) into (4): a^3 + 1/a^3 = 13*4 → 52
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