Let x = √5 + √3. Find the exact simplified value of: x + 1/x Give your answer in surd (radical) form, without decimal approximation.

Introduction / Context:
This question tests rationalization of surds. When x is a sum of square roots, 1/x can be simplified by multiplying numerator and denominator by the conjugate. After that, adding x + 1/x becomes straightforward.

Given Data / Assumptions:

• x = √5 + √3 • Required: x + 1/x • Exact surd form (no decimals)

Concept / Approach:
Use the conjugate: 1/(√5 + √3) = (√5 − √3) / ((√5 + √3)(√5 − √3)). The denominator becomes 5 − 3 = 2, eliminating radicals from the denominator. Then add the resulting expression to x.

Step-by-Step Solution:
1) Start with: x = √5 + √3 2) Compute 1/x by rationalizing: 1/x = 1/(√5 + √3) * (√5 − √3)/(√5 − √3) 3) Denominator: (√5 + √3)(√5 − √3) = 5 − 3 = 2 4) Therefore: 1/x = (√5 − √3)/2 5) Now add x + 1/x: x + 1/x = (√5 + √3) + (√5 − √3)/2 6) Write with common denominator 2: = (2√5 + 2√3 + √5 − √3)/2 7) Combine like terms: = (3√5 + √3)/2

Verification / Alternative check:
Approximate check: √5 ≈ 2.236, √3 ≈ 1.732, so x ≈ 3.968. Then 1/x ≈ 0.252. Sum ≈ 4.220. Our exact result: (3√5 + √3)/2 ≈ (3*2.236 + 1.732)/2 = (6.708 + 1.732)/2 = 4.220. Matches well.

Why Other Options Are Wrong:
• 2√15, 3√5, √15: these do not match the rationalized sum and give different approximations. • (√5 + 3√3)/2 swaps coefficients incorrectly.

Common Pitfalls:
• Forgetting to use the conjugate (√5 − √3). • Incorrectly multiplying and not getting 5 − 3 = 2.

Final Answer:
(3√5 + √3) / 2