The sum of the cubes of two given numbers is 10234, while the sum of the two given numbers is 34. What is the positive difference between the cubes of the two given numbers?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 3484.
a + b = 34, a³ + b³ = 10234
Use identity:
a³ + b³ = (a + b)³ − 3ab(a + b)
⇒ 10234 = 39304 − 102ab ⇒ ab = 285
Now, a³ − b³ = (a − b)(a² + ab + b²)
a² + b² = 34² − 2×285 = 1156 − 570 = 586
So, a² + ab + b² = 586 + 285 = 871
Find a − b:
(a + b)² − (a − b)² = 4ab ⇒ 1156 − (a − b)² = 1140 ⇒ (a − b)² = 16 ⇒ a − b = ±4
Therefore, |a³ − b³| = 4 × 871 = 3484