31³ + 18³−37³ + 210 is equal to:

Correct — B. Cube each number, then combine.

31³ = 961 × 31 = 29791
18³ = 324 × 18 = 5832
37³ = 1369 × 37 = 50653

Add the two positive cubes: 29791 + 5832 = 35623
Subtract 37³: 35623 − 50653 = −15030
Add 210: −15030 + 210 = −14820 → option (b)

Why the others are wrong

• (a)
  −36810 — −36810 is about 2.5 times the true size. 37³ ≈ 50,650 against 31³ + 18³ ≈ 35,620 puts the result near −15,000, nowhere near −36,810.
• (c)
  −45670 — −45670 is below −44611, the value of 18³ − 37³ + 210 with 31³ left out. Adding the positive 31³ can only raise the total, so it cannot sink this low.
• (d)
  −23450 — −23450 is below −20652, the value of 31³ − 37³ + 210 with 18³ left out. Adding the positive 18³ = 5832 can only raise the total above that.

This is direct computation with cubes of two-digit numbers. Building each cube as square × number (37² = 1369, then × 37) keeps it manageable.

Estimate before you multiply: 37³ ≈ 50,650 and 31³ + 18³ ≈ 35,620, so the answer is near −15,000. That alone separates the options.

The shortcut a³ + b³ + c³ = 3abc applies when a + b + c = 0. Here 31 + 18 − 37 = 12, so it does not apply.

Key facts

• 31³ = 29791, 18³ = 5832 and 37³ = 50653.
• a³ − b³ = (a − b)(a² + ab + b²).
• If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Common traps

• Applying a³ + b³ + c³ = 3abc without checking that a + b + c = 0. Here the sum is 12, and the shortcut would give 3 × 31 × 18 × (−37) = −61938, which is wrong.
• Dropping the + 210 at the end, which leaves −15030.

No directly related past PYQ was found.