Rationalize a reciprocal of radicals: Compute 1 / (√9 − √8) and express it in simplest surd form.
Aptitude
Square Root and Cube Root
Difficulty: Easy
Choose an option
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A(3 - 2√2) / 2
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B1 / (3 + 2√2)
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C(3 - 2√2)
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D(3 + 2√2)
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E2 / (3 - 2√2)
Answer
Correct Answer: (3 + 2√2)
Explanation
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:
1 / (3 − 2√2) * (3 + 2√2)/(3 + 2√2) = (3 + 2√2) / (9 − (2√2)^2).(2√2)^2 = 4 * 2 = 8.Denominator = 9 − 8 = 1.Therefore, the expression simplifies to 3 + 2√2.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)