To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we start by reducing the coefficients and the square roots.
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Simplify the coefficients: \[ \frac{12}{3} = 4 \]
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Simplify the square roots:
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For \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \]
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For \(\sqrt{300}\): \[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3} \]
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Substituting back into the expression: \[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{4 \cdot 2\sqrt{6}}{10\sqrt{3}} = \frac{8\sqrt{6}}{10\sqrt{3}} \]
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Simplify the fraction: \[ \frac{8}{10} = \frac{4}{5} \] Thus, \[ \frac{8\sqrt{6}}{10\sqrt{3}} = \frac{4\sqrt{6}}{5\sqrt{3}} \]
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Rationalize the denominator: To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\): \[ \frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{18}}{5 \cdot 3} = \frac{4\sqrt{18}}{15} \]
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Simplify the square root in the numerator: \[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \] Therefore, substituting back gives: \[ \frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15} \]
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Final simplification: \[ \frac{12}{15} = \frac{4}{5} \] Thus, final expression is: \[ \frac{4\sqrt{2}}{5} \]
The final answer in simplest radical form is: \[ \frac{4\sqrt{2}}{5} \]