If x = 2 + √3, y = 2 - √3 and z = 1, then what is the value of (x / (y z)) + (y / (x z)) + (z / (x y)) + 2[(1 / x) + (1 / y) + (1 / z)]?

Introduction / Context:
This question is another example of an expression built from conjugate surds, in this case 2 + √3 and 2 - √3. Such pairs often yield neat simplifications when multiplied, because many radical parts cancel out. Here, the expression combines several ratios and reciprocals involving x, y and z, and you are asked to evaluate the entire expression exactly. Recognising conjugate relationships and simplifying stepwise is the key to solving it efficiently.

Given Data / Assumptions:

• x = 2 + √3.
• y = 2 - √3.
• z = 1.
• We must evaluate E = x / (y z) + y / (x z) + z / (x y) + 2[(1 / x) + (1 / y) + (1 / z)].
• All denominators are nonzero and all roots are real.

Concept / Approach:
First, note that z = 1 simplifies many terms, since y z = y and x z = x, and 1 / z = 1. Also, the product x y for conjugates 2 ± √3 takes a particularly simple form. Once x y is computed, the term z / (x y) and the reciprocals 1 / x and 1 / y can be simplified by rationalising denominators using the conjugate pair. Then we can group terms cleverly to avoid long calculations and exploit symmetry, similarly to other problems involving conjugates.

Step-by-Step Solution:
Step 1: Compute x y = (2 + √3)(2 - √3) = 2^2 - (√3)^2 = 4 - 3 = 1. Step 2: Because z = 1, we have x / (y z) = x / y and y / (x z) = y / x. Step 3: Compute x / y = (2 + √3) / (2 - √3). Rationalising by multiplying numerator and denominator by (2 + √3) yields (2 + √3)^2 / (4 - 3) = (4 + 4√3 + 3) / 1 = 7 + 4√3. Step 4: Similarly, y / x = (2 - √3) / (2 + √3) = (2 - √3)^2 / (4 - 3) = (4 - 4√3 + 3) / 1 = 7 - 4√3. Step 5: Add x / y and y / x: (7 + 4√3) + (7 - 4√3) = 14. Step 6: Compute z / (x y). Since x y = 1 and z = 1, this equals 1 / 1 = 1. Step 7: Now compute 1 / x and 1 / y. Because x y = 1, we already know that 1 / x = y and 1 / y = x. So 1 / x + 1 / y + 1 / z = y + x + 1 = (2 - √3) + (2 + √3) + 1 = 5. Step 8: The given expression is E = (x / y + y / x + z / (x y)) + 2(1 / x + 1 / y + 1 / z) = (14 + 1) + 2 × 5 = 15 + 10 = 25.

Verification / Alternative check:
You can confirm the result numerically by approximating √3 as about 1.732, giving x ≈ 3.732 and y ≈ 0.268. Substituting these approximations into the expression and computing each term carefully will yield a sum very close to 25, confirming that the algebraic simplification is correct. This numerical check is reassuring when dealing with many surd manipulations.

Why Other Options Are Wrong:

• Options b (22), c (17), d (43) and e (19) arise from errors such as miscomputing x y, forgetting that 1 / x equals y, or miscounting the contribution of the term 2[(1 / x) + (1 / y) + (1 / z)].
• None of these values fit both the exact algebraic relationships and a careful numerical approximation.

Common Pitfalls:
A major pitfall is failing to notice that x and y are conjugates and that their product equals 1. Without this observation, you might perform unnecessary and error prone calculations for reciprocals. Another frequent mistake is mishandling the factor of 2 in front of the bracket or misinterpreting z = 1 when simplifying denominators. Always look for conjugate relationships and simple products before starting calculations, and check coefficients and constants carefully when combining terms.

Final Answer:
The value of the given expression is 25.