To simplify the expression \(\frac{12 \sqrt{24}}{3 \sqrt{30}}\), we can start by simplifying both the numerator and denominator step-by-step.
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Simplify the coefficients: \[ \frac{12}{3} = 4 \]
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Simplify the square roots: For \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \]
And for \(\sqrt{30}\), it cannot be simplified further because it has no perfect square factors.
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Replace in the expression: Now substitute back: \[ \frac{12 \sqrt{24}}{3 \sqrt{30}} = \frac{4 \cdot 2\sqrt{6}}{\sqrt{30}} = \frac{8\sqrt{6}}{\sqrt{30}} \]
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Rationalize the denominator: To rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{30}\): \[ \frac{8\sqrt{6} \cdot \sqrt{30}}{\sqrt{30} \cdot \sqrt{30}} = \frac{8\sqrt{180}}{30} \]
Simplifying \(\sqrt{180}\) gives: \[ \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5} \]
Therefore, substitute this back into the fraction: \[ \frac{8 \times 6 \sqrt{5}}{30} = \frac{48 \sqrt{5}}{30} \]
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Reduce the fraction: The greatest common divisor of 48 and 30 is 6, so simplify: \[ \frac{48 \div 6 \sqrt{5}}{30 \div 6} = \frac{8\sqrt{5}}{5} \]
Thus, the expression in simplest radical form is: \[ \frac{8\sqrt{5}}{5} \]