If a = √2 + 1 and b = √2 − 1 in basic surd manipulation, what is the exact simplified value of the expression 1/(a + 1) + 1/(b + 1)?

Introduction / Context:
This question checks understanding of surds, rationalisation, and algebraic simplification using conjugate based ideas. Such questions are common in aptitude exams to test manipulation of radicals without a calculator.

Given Data / Assumptions:

• a = √2 + 1
• b = √2 − 1
• Required expression is 1/(a + 1) + 1/(b + 1)
• All quantities are real numbers

Concept / Approach:
We simplify each denominator first, then add the two fractions. The key idea is that √2 + 1 and √2 − 1 are conjugate like expressions. Adding 1 to each makes simple surds that can be handled directly. We then compute a common denominator and simplify carefully. No special formulas beyond basic fraction addition are required.

Step-by-Step Solution:
a + 1 = (√2 + 1) + 1 = √2 + 2 b + 1 = (√2 − 1) + 1 = √2 So 1/(a + 1) = 1/(√2 + 2) And 1/(b + 1) = 1/√2 Required sum S = 1/(√2 + 2) + 1/√2 Take common denominator √2(√2 + 2) S = [√2 + (√2 + 2)] / [√2(√2 + 2)] S = [√2 + √2 + 2] / [√2(√2 + 2)] = [2√2 + 2] / [√2(√2 + 2)] Factor numerator: 2(√2 + 1) Denominator: √2(√2 + 2) = √2 * √2 + √2 * 2 = 2 + 2√2 = 2(1 + √2) So S = 2(√2 + 1) / [2(1 + √2)] = 1

Verification / Alternative check:
We can approximate numerically. √2 ≈ 1.414. Then a ≈ 2.414 and b ≈ 0.414. So a + 1 ≈ 3.414 and b + 1 ≈ 1.414. Then 1/(a + 1) ≈ 0.293 and 1/(b + 1) ≈ 0.707. Their sum is approximately 1.0 which matches the exact result.

Why Other Options Are Wrong:
2 is too large and would require each term to be roughly 1 which is not true. 0 would require the two fractions to cancel exactly which they do not. −1 contradicts the clearly positive nature of both fractions. 3 is far too large compared to the approximate numerical sum which is near 1.

Common Pitfalls:
Learners sometimes try to rationalise each surd separately, creating unnecessary steps. Another common mistake is using an incorrect common denominator while adding fractions with surd denominators. Some students also forget that both denominators are positive and incorrectly consider negative values.

Final Answer:
Therefore, the exact simplified value of 1/(a + 1) + 1/(b + 1) is 1.