Find the value of “n” where 512 + 1152 + 864 + 24n = (8 + 6)3.

Concept: The binomial cube identity (a + b)3 = a3 + 3a2b + 3ab2 + b3 expands the cube of a sum into four terms — the two cubes of the individual numbers and two middle terms carrying a factor of 3. Any numeric expansion written in this exact pattern can be matched term by term against a and b to find an unknown coefficient.

Application: Take a = 8 and b = 6, matching 512 + 1152 + 864 + 24n against the identity's four terms.

1. a3 = 83 = 512, matching the first term.

2. 3a2b = 3 × 82 × 6 = 1152, matching the second term.

3. 3ab2 = 3 × 8 × 62 = 864, matching the third term.

4. b3 = 63 = 216, which the expression represents as 24n — so 24n = 216.

5. Solving, n = 216 ÷ 24 = 9.

Cross-check: (8 + 6)3 = 143 = 2744, and 512 + 1152 + 864 + 216 = 2744 — the two independently computed totals agree, confirming n = 9.