Given √2 ≈ 1.4142, compute the decimal value of 7 / (4 + √2) to four decimal places by rationalizing the denominator.

Introduction / Context:Fractions with radicals in the denominator are best handled by rationalizing the denominator. This simplifies numerical evaluation and reduces rounding errors. Here, a short rationalization step converts the expression to a simple combination of integers and √2.

Given Data / Assumptions:

• Expression: 7 / (4 + √2), with √2 ≈ 1.4142.
• Accuracy requested: four decimal places (or as presented in the options).

Concept / Approach:Multiply numerator and denominator by the conjugate (4 − √2). The product in the denominator becomes a difference of squares, eliminating the radical. Then simplify and evaluate using the given approximation for √2.

Step-by-Step Solution:7 / (4 + √2) = 7(4 − √2) / [(4 + √2)(4 − √2)].Denominator: 16 − (√2)^2 = 16 − 2 = 14.Thus the value = 7(4 − √2) / 14 = (4 − √2) / 2.Compute numerically: (4 − 1.4142)/2 = 2.5858/2 = 1.2929.

Verification / Alternative check:Direct calculator check of 7/(4 + 1.4142) yields ≈ 1.2929, matching the rationalized computation.

Why Other Options Are Wrong:

• 3.5858 / 4.4142 / 1.5858 / 1.4142: These correspond to 4 − √2, 4 + √2, or incomplete division by 2, not the final simplified value.

Common Pitfalls:Forgetting to divide by 2 after rationalizing; using √2 ≈ 1.41 too early which can shift the thousandths place.

Final Answer:1.2929