To simplify the expression \(\frac{30\sqrt{14}}{6\sqrt{2}}\), you can start by simplifying the constants and the square roots separately:
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Simplify the constants: \[ \frac{30}{6} = 5 \]
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Simplify the square roots: You have \(\sqrt{14}\) and \(\sqrt{2}\). You can rewrite \(\sqrt{14}\) as \(\sqrt{7 \cdot 2}\) or \(\sqrt{14} = \sqrt{2} \cdot \sqrt{7}\).
Thus, the expression becomes: \[ \frac{30\sqrt{14}}{6\sqrt{2}} = 5 \cdot \frac{\sqrt{14}}{\sqrt{2}} \]
Using the property of square roots \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\): \[ \frac{\sqrt{14}}{\sqrt{2}} = \sqrt{\frac{14}{2}} = \sqrt{7} \]
So now, you can combine everything: \[ 5 \cdot \sqrt{7} = 5\sqrt{7} \]
Therefore, the simplified form of the expression \(\frac{30\sqrt{14}}{6\sqrt{2}}\) is: \[ \boxed{5\sqrt{7}} \]
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