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Step 1:I'll solve this step by step using the specified LaTeX formatting guidelines:
Step 2:: Identify the number under the square root
\sqrt{224}
Step 3:: Find the largest perfect square factor of 224
• $$224 = 2^{5} \times 7
• First, let's break down 224 into its prime factorization
Step 4:: Separate the perfect square factor
• $$\sqrt{224} = \sqrt{2^{4} \times 2 \times 7}
• \sqrt{224} = \sqrt{16 \times 2 \times 7}
Step 5:: Simplify the square root
• $$\sqrt{224} = \sqrt{16} \times \sqrt{2 \times 7}
• \sqrt{224} = 4 \times \sqrt{14}
Final Answer
Key insights: - Always look for the largest perfect square factor - Separate the square root into a coefficient and remaining radical - Simplify when possible
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