To simplify the expression \( \frac{18}{\sqrt{24}} \), we can start by simplifying the square root in the denominator.
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First, simplify \( \sqrt{24} \): \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \]
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Now substitute \( \sqrt{24} \) back into the expression: \[ \frac{18}{\sqrt{24}} = \frac{18}{2\sqrt{6}} \]
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Simplify the fraction: \[ \frac{18}{2\sqrt{6}} = \frac{9}{\sqrt{6}} \]
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To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{6} \): \[ \frac{9}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{9\sqrt{6}}{6} \]
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Finally, simplify the fraction: \[ \frac{9\sqrt{6}}{6} = \frac{3\sqrt{6}}{2} \]
Thus, the simplified form of \( \frac{18}{\sqrt{24}} \) is \( \frac{3\sqrt{6}}{2} \).