(sqrt (18)cdot sqrt (12))/(sqrt (24))

To simplify the expression \(\frac{\sqrt{18} \cdot \sqrt{12}}{\sqrt{24}}\), we can use the properties of square roots and simplify each part separately.<br /><br />First, let's simplify the numerator:<br />\[<br />\sqrt{18} \cdot \sqrt{12} = \sqrt{18 \cdot 12}<br />\]<br /><br />Next, calculate \(18 \cdot 12\):<br />\[<br />18 \cdot 12 = 216<br />\]<br /><br />So, the numerator becomes:<br />\[<br />\sqrt{216}<br />\]<br /><br />Now, let's simplify the denominator:<br />\[<br />\sqrt{24}<br />\]<br /><br />We need to simplify each square root. Start with \(\sqrt{216}\):<br />\[<br />216 = 2^3 \cdot 3^3<br />\]<br />\[<br />\sqrt{216} = \sqrt{2^3 \cdot 3^3} = \sqrt{(2 \cdot 3)^3} = \sqrt{6^3} = 6^{3/2} = 6 \sqrt{6}<br />\]<br /><br />Next, simplify \(\sqrt{24}\):<br />\[<br />24 = 2^3 \cdot 3<br />\]<br />\[<br />\sqrt{24} = \sqrt{2^3 \cdot 3} = \sqrt{(2^2 \cdot 2) \cdot 3} = \sqrt{4 \cdot 6} = 2 \sqrt{6}<br />\]<br /><br />Now, substitute these simplified forms back into the original expression:<br />\[<br />\frac{\sqrt{216}}{\sqrt{24}} = \frac{6 \sqrt{6}}{2 \sqrt{6}}<br />\]<br /><br />The \(\sqrt{6}\) terms cancel out:<br />\[<br />\frac{6 \sqrt{6}}{2 \sqrt{6}} = \frac{6}{2} = 3<br />\]<br /><br />Therefore, the simplified form of the expression is:<br />\[<br />\boxed{3}<br />\]