In algebra, two real numbers a and b satisfy a + b = 8 and a * b = 15. Using these conditions, determine the exact value of the sum of cubes a^3 + b^3.

Introduction / Context:
This question checks your ability to use algebraic identities to compute higher power expressions from simple symmetric sums. Instead of finding a and b explicitly, which would require solving a quadratic equation, you can use the identity for a^3 + b^3 in terms of a + b and a b. This approach is efficient and frequently used in algebra and aptitude questions involving polynomials.

Given Data / Assumptions:

• The two real numbers a and b satisfy a + b = 8.
• Their product is a * b = 15.
• We need to compute a^3 + b^3 exactly.
• Standard identities for powers and symmetric expressions apply.

Concept / Approach:
The key identity is (a + b)^3 = a^3 + b^3 + 3 a b (a + b). From this, we can solve for a^3 + b^3 as a^3 + b^3 = (a + b)^3 − 3 a b (a + b). Since both a + b and a b are given, this formula allows us to find a^3 + b^3 in one line of computation without solving for a and b individually. This is a standard technique to simplify problems involving symmetric expressions.

Step-by-Step Solution:
1) Recall the identity: (a + b)^3 = a^3 + b^3 + 3 a b (a + b). 2) Rearrange it to solve for a^3 + b^3: a^3 + b^3 = (a + b)^3 − 3 a b (a + b). 3) Substitute the given values a + b = 8 and a b = 15. 4) Compute (a + b)^3 = 8^3 = 512. 5) Compute 3 a b (a + b) = 3 * 15 * 8 = 45 * 8 = 360. 6) Now calculate a^3 + b^3 = 512 − 360 = 152.

Verification / Alternative check:
To verify, you can find the actual values of a and b by solving the quadratic equation t^2 − (a + b)t + a b = 0, which becomes t^2 − 8t + 15 = 0. Factoring gives (t − 3)(t − 5) = 0, so a and b are 3 and 5 in some order. Then compute a^3 + b^3 = 3^3 + 5^3 = 27 + 125 = 152. This matches the result obtained from the identity and confirms that the identity based method is correct.

Why Other Options Are Wrong:
Options b (124), c (98), d (260), and e (176) do not satisfy the identity when plugged into (a + b)^3 = a^3 + b^3 + 3 a b (a + b). For example, if a^3 + b^3 were 124, then (a + b)^3 − 3 a b (a + b) would not equal 124, given a + b = 8 and a b = 15. Only option a corresponds to the correct calculation of 512 − 360.

Common Pitfalls:
Some learners confuse the identity for a^3 + b^3 with that for (a + b)^2 or (a − b)^3, leading to missing or extra terms. Others attempt to cube a + b directly without subtracting 3 a b (a + b), which effectively gives (a + b)^3 by itself instead of a^3 + b^3. Mis calculating 8^3 or 3 * 15 * 8 is another source of error. Writing each step clearly and checking simple arithmetic helps avoid these mistakes.

Final Answer:
Using the cube identity, the exact value of a^3 + b^3 is 152.