31³ + 18³ - 37³ + 210 is equal to:

Question

31³ + 18³ - 37³ + 210 is equal to:

1. A. -36810
2. B. -14820 (Correct answer)
3. C. -45670
4. D. -23450

Correct Answer

Option B — -14820

Detailed Solution & Explanation

The correct answer is -14820.

Key Points

• Compute the cubes: 31³ = 29,791, 18³ = 5,832, 37³ = 50,653.
• Then 29,791 + 5,832 − 50,653 + 210 = 35,623 − 50,653 + 210 = −14,820.
• With no factoring shortcut available (the three bases don't sum to zero), evaluate each cube and combine left to right; the large negative 37³ term drives the total below zero.

Additional Information

• The identity worth checking first is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca), which collapses to 3abc when a + b + c = 0.
• Here the expression is 31³ + 18³ − 37³, i.e. a = 31, b = 18, c = −37. Since 31 + 18 − 37 = 12 ≠ 0, the shortcut does not apply and direct computation is required.
• Had the bases summed to zero — for example 31, 18 and −49 — the answer would have been 3abc immediately, which is the version examiners usually set.
• Cubes worth memorising to speed this up: 18³ = 5,832, 31³ = 29,791, 37³ = 50,653.