Given √3 ≈ 1.732 and √2 ≈ 1.414, compute the value of 1 / (√3 + √2). Provide the value to three decimal places.

Introduction / Context: This numerical approximation problem practices handling radicals and reciprocals. Rationalization gives an exact form, while given approximations yield the decimal result.

Given Data / Assumptions:

• √3 ≈ 1.732, √2 ≈ 1.414.
• Expression: 1 / (√3 + √2).
• Round to three decimal places.

Concept / Approach: Using the conjugate: 1 / (√3 + √2) = (√3 − √2) / [(√3 + √2)(√3 − √2)] = (√3 − √2) / (3 − 2) = √3 − √2. Then substitute the approximations.

Step-by-Step Solution: 1 / (√3 + √2) = √3 − √2. ≈ 1.732 − 1.414 = 0.318. To three decimals, the value is 0.318.

Verification / Alternative check: Direct division: 1 / (1.732 + 1.414) ≈ 1 / 3.146 ≈ 0.3179, which rounds to 0.318 and matches the conjugate method.

Why Other Options Are Wrong: 0.064 and 0.308 deviate from the rationalized exact form √3 − √2. 2.146 is vastly too large for a reciprocal of a number greater than 3.

Common Pitfalls: Forgetting to rationalize or incorrectly subtracting decimal approximations may produce small but significant errors. Always check with both methods if unsure.

Final Answer: 0.318