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Step 1:I'll solve this problem step by step using the specified LaTeX formatting guidelines:
Step 2:: Understand the Problem
The problem asks us to simplify the expression $$4 \sqrt{72}
Step 3:: Simplify the Radical
72 = 2^{3} \times 3^{2}
First, let's break down 72 into its prime factors:
Step 4:: Separate Perfect Square Factors
\sqrt{72} = \sqrt{2^{3} \times 3^{2}} = \sqrt{36 \times 2}
We want to pull out the largest perfect square from under the radical:
Step 5:: Simplify the Radical
\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6 \times \sqrt{2}
Step 6:: Multiply by the Original Coefficient
4 \sqrt{72} = 4 \times 6 \times \sqrt{2} = 24\sqrt{2}
Final Answer
24\sqrt{2}
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