Use algebraic identities to avoid expanding fully when possible.\nIf a - b = 2 and ab = 8, what is the value of a^3 - b^3?

Introduction / Context:
This question tests the use of algebraic identities and relationships between a and b. Instead of solving for a and b explicitly, we can compute a^3 - b^3 using the factorization a^3 - b^3 = (a - b)(a^2 + ab + b^2). We are given a - b and ab, so we just need a^2 + b^2.

Given Data / Assumptions:

• a - b = 2
• ab = 8
• We need a^3 - b^3

Concept / Approach:
Use identities:\n1) a^3 - b^3 = (a - b)(a^2 + ab + b^2)\n2) (a - b)^2 = a^2 - 2ab + b^2, so a^2 + b^2 = (a - b)^2 + 2ab.\nThen substitute known values and multiply.

Step-by-Step Solution:
Use: a^3 - b^3 = (a - b)(a^2 + ab + b^2) We know (a - b) = 2 and ab = 8, so we need a^2 + b^2 Compute a^2 + b^2 from (a - b)^2: (a - b)^2 = a^2 - 2ab + b^2 So a^2 + b^2 = (a - b)^2 + 2ab = 2^2 + 2*8 = 4 + 16 = 20 Now a^2 + ab + b^2 = (a^2 + b^2) + ab = 20 + 8 = 28 Finally: a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 2 * 28 = 56

Verification / Alternative check:
You could solve a - b = 2 and ab = 8 by setting a = b + 2, giving b(b+2)=8 => b^2+2b-8=0, so b=2 or b=-4 and a=4 or a=-2. In both cases, a^3 - b^3 = 56, confirming the identity-based result.

Why Other Options Are Wrong:
34 and 43 come from incorrect computation of a^2 + b^2 or forgetting the +ab term.
40 is a common result if you use (a - b)(a^2 + b^2) and omit ab.
65 is unrelated and typically comes from random expansion mistakes.

Common Pitfalls:
Forgetting that a^2 + ab + b^2 is needed (not just a^2 + b^2), and misusing (a - b)^2 as a^2 - b^2. Also, sign errors with 2ab are common.

Final Answer:
a^3 - b^3 = 56