Let x = 2 + √3 and y = 2 − √3. Find the exact value of (x^2 + y^2) / (x^3 + y^3).

Introduction / Context:
This problem involves conjugate surds x = 2 + √3 and y = 2 − √3. Sums and products of such pairs often simplify nicely because cross terms cancel. The task is to evaluate the ratio (x^2 + y^2) / (x^3 + y^3) exactly.

Given Data / Assumptions:

• x = 2 + √3 and y = 2 − √3
• We need x^2 + y^2 and x^3 + y^3
• Use identities with conjugates to simplify.

Concept / Approach:
Use symmetric identities: x + y and xy are rational for conjugates. Specifically, (x + y) = 4 and xy = (2 + √3)(2 − √3) = 4 − 3 = 1. Then apply formulas: x^2 + y^2 = (x + y)^2 − 2xy and x^3 + y^3 = (x + y)^3 − 3xy(x + y).

Step-by-Step Solution:
Compute x + y = (2 + √3) + (2 − √3) = 4.Compute xy = (2 + √3)(2 − √3) = 4 − 3 = 1.x^2 + y^2 = (x + y)^2 − 2xy = 4^2 − 2*1 = 16 − 2 = 14.x^3 + y^3 = (x + y)^3 − 3xy(x + y) = 4^3 − 3*1*4 = 64 − 12 = 52.Therefore, (x^2 + y^2)/(x^3 + y^3) = 14/52 = 7/26.

Verification / Alternative check:
Numerically, x ≈ 3.732 and y ≈ 0.268. Compute x^2 + y^2 ≈ 13.93 + 0.072 ≈ 14.002 and x^3 + y^3 ≈ 51.96 + 0.036 ≈ 51.996 → ratio ≈ 0.2692 ≈ 7/26 (≈ 0.26923).

Why Other Options Are Wrong:
7/38, 7/40, and 7/19 come from incorrect substitution or algebra. 1/2 is a common guess but does not match the exact identity-based result.

Common Pitfalls:
Forgetting identities for sums of squares and cubes; mixing up xy and x + y; or approximating too early and losing exactness. Work symbolically first, then check numerically.

Final Answer:
7/26