If a = 2 and b = 3, evaluate the expression: 27a^3 − 54a^2b + 36ab^2 − 8b^3 Simplify using a suitable algebraic identity and compute the exact value.

Introduction / Context:
This problem tests recognition of a binomial cube pattern hidden inside a longer polynomial. Expressions like 27a^3 − 54a^2b + 36ab^2 − 8b^3 are not meant to be expanded or computed term-by-term unless necessary. Instead, you should notice the structure matches (3a − 2b)^3. Recognizing identities is a key skill in simplification questions because it reduces time and prevents arithmetic mistakes. Once identified, the expression becomes a simple cube of a linear term, and substituting a = 2 and b = 3 becomes trivial. Many wrong answers come from plugging in values directly and making large-number calculation errors, or from misreading signs. The identity approach is faster and more reliable. Also, because 3a and 2b become equal for the given values, the inner term becomes zero, which makes the entire cube zero. That makes this question a classic “collapse to zero” pattern.

Given Data / Assumptions:

• a = 2 • b = 3 • Expression: 27a^3 − 54a^2b + 36ab^2 − 8b^3 • Identity: (p − q)^3 = p^3 − 3p^2q + 3pq^2 − q^3

Concept / Approach:
Compare the given expression to (p − q)^3. If we set p = 3a and q = 2b, then: p^3 = 27a^3, −3p^2q = −3*(9a^2)*(2b) = −54a^2b, 3pq^2 = 3*(3a)*(4b^2) = 36ab^2, −q^3 = −8b^3. So the whole expression equals (3a − 2b)^3. Then substitute the given values and evaluate.

Step-by-Step Solution:
1) Identify p = 3a and q = 2b. 2) Recognize the expression as: (3a − 2b)^3 3) Substitute a = 2 and b = 3 into the inner term: 3a − 2b = 3*2 − 2*3 = 6 − 6 = 0 4) Cube it: (3a − 2b)^3 = 0^3 = 0

Verification / Alternative check:
Direct substitution check (conceptually): since 3a equals 2b for these values, the polynomial that represents (3a − 2b)^3 must be zero. Any correct computation method must yield 0. This makes the identity-based reasoning very robust and confirms the result without long arithmetic.

Why Other Options Are Wrong:
• 1562, 1616, 1676, 1728: these arise from incorrect direct term-by-term substitution, sign errors, or treating the expression as (3a + 2b)^3. • If someone uses plus instead of minus, the result becomes (3a + 2b)^3 = 12^3 = 1728, explaining that distractor.

Common Pitfalls:
• Missing the identity and doing long calculations. • Confusing (p − q)^3 with (p + q)^3. • Sign mistakes in the middle terms (−54 and +36).

Final Answer:
0