Given √2 ≈ 1.4142, compute the value of 7 / (3 + √2). Provide the result to four decimal places.

Introduction / Context: This is a straightforward evaluation of a rational expression involving a radical in the denominator. Rationalization provides insight; approximation provides a quick decimal answer.

Given Data / Assumptions:

• √2 ≈ 1.4142.
• Expression: 7 / (3 + √2).
• Round to four decimal places.

Concept / Approach: Either directly substitute √2 and compute, or rationalize by multiplying numerator and denominator by (3 − √2) to check the magnitude. Both lead to the same decimal result.

Step-by-Step Solution: Direct evaluation: denominator = 3 + 1.4142 = 4.4142. Value = 7 / 4.4142 ≈ 1.5858.

Verification / Alternative check: Rationalize: 7 / (3 + √2) = 7(3 − √2) / (9 − 2) = 7(3 − √2) / 7 = 3 − √2 ≈ 3 − 1.4142 = 1.5858, confirming the result.

Why Other Options Are Wrong: 4.4142 is the denominator, not the value. 3.4852 and 3.5858 are unrelated approximations and not equal to 3 − √2.

Common Pitfalls: Confusing the denominator with the final result, or forgetting to round consistently.

Final Answer: 1.5858