To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we will first simplify both the numerator and the denominator.
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Simplify the numerator \(12\sqrt{24}\):
We can simplify \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \] Now substituting back into the numerator: \[ 12\sqrt{24} = 12 \cdot 2\sqrt{6} = 24\sqrt{6} \]
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Simplify the denominator \(3\sqrt{300}\):
We can simplify \(\sqrt{300}\): \[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3} \] Now substituting back into the denominator: \[ 3\sqrt{300} = 3 \cdot 10\sqrt{3} = 30\sqrt{3} \]
Now we can rewrite the expression: \[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}} \]
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Simplify the fraction:
We can simplify the coefficients: \[ \frac{24}{30} = \frac{4}{5} \] We can also rewrite the whole expression: \[ \frac{24\sqrt{6}}{30\sqrt{3}} = \frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} = \frac{4}{5} \cdot \sqrt{\frac{6}{3}} = \frac{4}{5} \cdot \sqrt{2} \]
Putting it all together, we get: \[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{4\sqrt{2}}{5} \]
Thus, the quotient in simplest radical form is:
\[ \boxed{\frac{4\sqrt{2}}{5}} \]