Rationalize a reciprocal of radicals: Compute 1 / (√9 − √8) and express it in simplest surd form.

Introduction / Context:Rationalizing denominators with surds is a classic skill. Using conjugates eliminates radicals in the denominator and yields a clean exact expression.

Given Data / Assumptions:

• Expression: 1 / (√9 − √8) = 1 / (3 − 2√2).
• We use the conjugate (3 + 2√2).

Concept / Approach:Multiply numerator and denominator by the conjugate to create a difference of squares in the denominator.

Step-by-Step Solution:

Verification / Alternative check:Numerical check: 3 + 2√2 ≈ 3 + 2*1.414 = 5.828; 1/(3 − 2*1.414) ≈ 1/(0.172) ≈ 5.814 (difference due to rounding of √2); exact values match symbolically.

Why Other Options Are Wrong:

• (3 − 2√2)/2 and (3 − 2√2) are not equal to the simplified reciprocal.
• 1/(3 + 2√2) is the reciprocal of the correct answer.

Common Pitfalls:Forgetting that (a − b)(a + b) = a^2 − b^2 and mis-squaring 2√2.

Final Answer:(3 + 2√2)