If a = √2 + 1 and b = √2 - 1, then what is the exact value of the expression 1/(a + 1) + 1/(b + 1)?

Introduction / Context:
This question tests simplification of expressions involving surds (square roots) and rationalization style thinking. The numbers √2 + 1 and √2 - 1 are conjugate like numbers, and they often lead to simple results when combined in symmetric expressions.

Given Data / Assumptions:

• a = √2 + 1.
• b = √2 - 1.
• We need the value of 1/(a + 1) + 1/(b + 1).
• All quantities are real and denominators are non zero.

Concept / Approach:
We first simplify a + 1 and b + 1, then compute their reciprocals. Because a and b are closely related, the resulting fractions combine in a neat way. Recognizing conjugate structures and simplifying step by step is the main idea.

Step-by-Step Solution:
Step 1: Compute a + 1: a + 1 = (√2 + 1) + 1 = √2 + 2. Step 2: Compute b + 1: b + 1 = (√2 - 1) + 1 = √2. Step 3: Find 1/(a + 1) = 1/(√2 + 2). Step 4: Find 1/(b + 1) = 1/√2. Step 5: Add the two fractions: 1/(√2 + 2) + 1/√2. Step 6: A quick method is to rationalize both or use a symbolic simplification. One systematic way is to simplify numerically and then recognize the exact form. Step 7: After simplification, the sum evaluates exactly to 1.

Verification / Alternative check:
We can rationalize 1/(√2 + 2) by multiplying numerator and denominator by (√2 - 2). However, even without full symbolic work, substituting approximate values √2 ≈ 1.414 gives a + 1 ≈ 3.414 and b + 1 ≈ 1.414. Then 1/3.414 ≈ 0.293 and 1/1.414 ≈ 0.707, whose sum is approximately 1.000, strongly suggesting the exact answer is 1.

Why Other Options Are Wrong:
Option A (0) would require the two terms to cancel exactly, which is impossible here. Option C (2) and Option D (3) are much larger than the computed numerical value. Option E (-1) is negative and does not match the clearly positive sum of two positive fractions.

Common Pitfalls:
A typical error is to confuse a and b with their inverses, or to try to rationalize in a complicated way and make algebraic slips. Some learners also miscalculate √2 and arrive at approximate sums that are slightly away from 1. Careful arithmetic or symbolic simplification avoids these mistakes.

Final Answer:

The exact value of the expression is 1.