Answer
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Step 1:I'll solve this step-by-step using the specified LaTeX formatting guidelines:
Step 2:: Identify the components of the radical
- We have $$\sqrt{24x^{5}}
- This involves simplifying a square root with a coefficient and a variable
Step 3:: Factor out perfect square terms from under the radical
- First, factor 24: $$24 = 4 \times 6
- Factor the variable: x^{5} = x^{4} \times x
Step 4:: Rewrite the expression
\sqrt{24x^{5}} = \sqrt{4 \times 6 \times x^{4} \times x}
Step 5:: Separate perfect square terms
= \sqrt{4} \times \sqrt{6} \times \sqrt{x^{4}} \times \sqrt{x}
Step 6:: Simplify each term
- $$\sqrt{4} = 2
- \sqrt{x^{4}} = x^{2}
Step 7:: Combine simplified terms
= 2 \times \sqrt{6} \times x^{2} \times \sqrt{x}
Step 8:: Final simplification
= 2x^{2}\sqrt{6x}
Final Answer
2x^{2}\sqrt{6x}
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