To simplify the expression \(\frac{9 \sqrt{80}}{4 \sqrt{90}}\), let's begin by simplifying both the numerator and the denominator.
Step 1: Simplify the square roots.
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Simplifying \(\sqrt{80}\): \[ \sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5} \]
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Simplifying \(\sqrt{90}\): \[ \sqrt{90} = \sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3 \sqrt{10} \]
Step 2: Substitute the simplified square roots back into the expression. Now, substitute back into the original expression: \[ \frac{9 \sqrt{80}}{4 \sqrt{90}} = \frac{9 \cdot 4 \sqrt{5}}{4 \cdot 3 \sqrt{10}} \]
Step 3: Cancel the \(4\) in the numerator and denominator: \[ = \frac{9 \sqrt{5}}{3 \sqrt{10}} \]
Step 4: Simplify the fraction: Now, divide both numerator and denominator by \(3\): \[ = \frac{3 \sqrt{5}}{\sqrt{10}} \]
Step 5: Rationalize the denominator: To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{10}\): \[ = \frac{3 \sqrt{5} \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} = \frac{3 \sqrt{50}}{10} \]
Step 6: Simplify \(\sqrt{50}\): \[ \sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2} \] So substitute back: \[ = \frac{3 \cdot 5 \sqrt{2}}{10} = \frac{15 \sqrt{2}}{10} \]
Step 7: Simplify the fraction: Dividing both the numerator and the denominator by \(5\): \[ = \frac{3 \sqrt{2}}{2} \]
Thus, the final answer in simplest form is: \[ \frac{3 \sqrt{2}}{2} \]