If a - b = 4 and the product ab = 45, then what is the value of a^3 - b^3?

Introduction / Context:

This algebra question focuses on using standard identities to avoid solving for a and b directly. You are given the difference a - b and the product ab and asked to find a^3 - b^3. Using algebraic identities is much faster and cleaner than trying to find the actual values of a and b from scratch.

Given Data / Assumptions:

• a - b = 4.
• ab = 45.
• We need to calculate a^3 - b^3.
• a and b are real numbers.

Concept / Approach:

Use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2). The given values provide a - b and ab directly. To apply the identity, we must compute a^2 + ab + b^2, which can be derived using the square of a - b.

Step-by-Step Solution:

Verification / Alternative check:

You can check by solving for a and b explicitly. The quadratic t^2 - (a + b)t + ab = 0 can be built once a + b is known. From (a - b)^2 = a^2 + b^2 - 2ab and the earlier calculations, you can find a and b numerically and verify that their cubes differ by 604.

Why Other Options Are Wrong:

The values 370, 253, 199, and 144 correspond to partial or incorrect use of identities. For example, forgetting the ab term or using a^2 + b^2 instead of a^2 + ab + b^2 will lead to one of these wrong numbers.

Common Pitfalls:

Students sometimes confuse the identity for a^3 - b^3 with the identity for a^3 + b^3 or forget to compute a^2 + b^2 correctly. Another common error is to attempt long and messy substitution instead of using the identities in a systematic way.

Final Answer:

The value of a^3 - b^3 is 604.