CDS 2024 · PYQ · Algebra / Cube Roots · hard
If p = ∛(a + √(a² + b³)) + ∛(a − √(a² + b³)), then what is p³ + 3bp equal to ?
- A.−2a
- B.a
- C.2a✓ Correct
- D.3a
Let u = ∛(a + √(a²+b³)) and v = ∛(a − √(a²+b³)). Then p = u + v. u³ + v³ = 2a and uv = ∛((a)² − (a²+b³)) = ∛(−b³) = −b. Using (u+v)³ = u³ + v³ + 3uv(u+v): p³ = 2a + 3(−b)p = 2a − 3bp. So p³ + 3bp = 2a.
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