To simplify the expression \(\frac{15 \sqrt{90}}{\sqrt{125}}\), follow these steps:
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Simplify the square roots:
- \(\sqrt{90} = \sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3\sqrt{10}\)
- \(\sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}\)
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Substitute back into the expression: \[ \frac{15 \cdot 3\sqrt{10}}{5\sqrt{5}} = \frac{45\sqrt{10}}{5\sqrt{5}} \]
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Simplify the coefficients: \[ \frac{45}{5} = 9 \] So the expression now is: \[ 9 \cdot \frac{\sqrt{10}}{\sqrt{5}} = 9 \cdot \sqrt{\frac{10}{5}} = 9 \cdot \sqrt{2} \]
Therefore, the simplified expression is: \[ 9\sqrt{2} \]