The sum of the squares of two real numbers is 41, and the sum of the two numbers themselves is 9. What is the sum of the cubes of these two numbers?

Introduction / Context:
This problem combines algebraic identities with simple equation solving. We are given the sum of two real numbers and the sum of their squares, and we are asked to find the sum of their cubes. Instead of solving directly for each number, we can use standard algebraic identities, which is a typical technique required in aptitude and entrance examinations.

Given Data / Assumptions:

• Let the numbers be a and b.
• a + b = 9.
• a^2 + b^2 = 41.
• We need to compute a^3 + b^3.
• Both a and b are real numbers.

Concept / Approach:
We use two key identities. First, (a + b)^2 = a^2 + b^2 + 2ab, which allows us to find the product ab from the given sums. Second, a^3 + b^3 can be expressed as (a + b)^3 - 3ab(a + b). Once we find ab, we substitute into this identity and compute the desired sum directly, avoiding the need to find a and b individually.

Step-by-Step Solution:
Given a + b = 9 and a^2 + b^2 = 41.Use the identity (a + b)^2 = a^2 + b^2 + 2ab.Compute (a + b)^2 = 9^2 = 81.Set up the equation: 81 = 41 + 2ab.Rearrange to get 2ab = 81 - 41 = 40.So ab = 40 / 2 = 20.Now use the identity for cubes: a^3 + b^3 = (a + b)^3 - 3ab(a + b).Compute (a + b)^3 = 9^3 = 729.Compute 3ab(a + b) = 3 * 20 * 9 = 540.So a^3 + b^3 = 729 - 540 = 189.

Verification / Alternative check:
An alternative is to solve for a and b as roots of the quadratic t^2 - 9t + 20 = 0, derived from t^2 - (a + b)t + ab = 0.This gives t = 4 and t = 5, so the numbers are 4 and 5.Compute their cubes: 4^3 = 64 and 5^3 = 125.Then a^3 + b^3 = 64 + 125 = 189, which confirms the earlier result.

Why Other Options Are Wrong:
169, 209, and 198 do not equal (a + b)^3 - 3ab(a + b) when a + b = 9 and ab = 20.If we mistakenly used incorrect identities, we might arrive at these distractor values, but direct verification with a = 4 and b = 5 shows that only 189 is correct.

Common Pitfalls:
A frequent mistake is forgetting the factor 3 in the identity a^3 + b^3 = (a + b)^3 - 3ab(a + b).Some learners also confuse the identity with that for (a - b)^3 or try to expand the cubes directly, which is slower and more error prone.Another pitfall is miscomputing (a + b)^2 or (a + b)^3, which leads to incorrect values of ab or the final sum.

Final Answer:
The sum of the cubes of the two numbers is 189.