To simplify the expression \(\frac{8\sqrt{27}}{2\sqrt{45}}\), we'll start by simplifying both the numerator and the denominator.
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Simplify \(\sqrt{27}\): \[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \]
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Simplify \(\sqrt{45}\): \[ \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]
Now, substituting back into the expression, we have: \[ \frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8 \cdot 3\sqrt{3}}{2 \cdot 3\sqrt{5}} \]
This simplifies further: \[ = \frac{24\sqrt{3}}{6\sqrt{5}} \]
Next, we simplify the fraction by dividing both the numerator and denominator by 6: \[ = \frac{4\sqrt{3}}{\sqrt{5}} \]
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{5}\): \[ = \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5} \]
Thus, the expression \(\frac{8\sqrt{27}}{2\sqrt{45}}\) simplifies to: \[ \boxed{\frac{4\sqrt{15}}{5}} \]