Simplify a radical expression by factoring: Evaluate (√24 + √216) / √96.
Aptitude
Square Root and Cube Root
Difficulty: Easy
Choose an option
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A2√6
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B6√2
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C2
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D2/√6
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E√6
Answer
Correct Answer: 2
Explanation
Introduction / Context:This expression simplifies elegantly by factoring out common square-free parts from each square root. Recognizing multiples of 6 under the radicals is the key.
Given Data / Assumptions:
- Expression: (√24 + √216) / √96.
- All radicals are positive principal roots.
Concept / Approach:Rewrite each radicand as a product of a perfect square and a square-free part, then cancel common factors between numerator and denominator.
Step-by-Step Solution:
√24 = √(4 * 6) = 2√6.√216 = √(36 * 6) = 6√6.Numerator = 2√6 + 6√6 = 8√6.√96 = √(16 * 6) = 4√6.Therefore, (√24 + √216) / √96 = (8√6) / (4√6) = 2.Verification / Alternative check:Numeric check: √24 ≈ 4.899, √216 ≈ 14.697, sum ≈ 19.596; √96 ≈ 9.798; ratio ≈ 2.00.
Why Other Options Are Wrong:
- 2√6, 6√2, 2/√6, √6 do not equal the simplified value of the ratio.
Common Pitfalls:Forgetting to factor out the perfect squares or failing to cancel √6 correctly.
Final Answer:2