To simplify the expression \( \frac{9}{4} \times \frac{49}{25} \div 7 \div 3 \times 4 \times 5 \times 5 \div 7 \div 3 \), we can break it down step by step.
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First, convert the division into multiplication by the reciprocal: \[ \frac{9}{4} \times \frac{49}{25} \times \frac{1}{7} \times \frac{1}{3} \times 4 \times 5 \times 5 \times \frac{1}{7} \times \frac{1}{3} \]
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Combine all factors: \[ = \frac{9 \times 49 \times 4 \times 5 \times 5}{4 \times 25 \times 7 \times 3 \times 7 \times 3} \]
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Simplify \(4\) and \(4\) in the numerator and denominator: \[ = \frac{9 \times 49 \times 5 \times 5}{25 \times 7 \times 3 \times 7 \times 3} \]
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\(49 = 7^2\), so we can rewrite the numerator and the denominator: \[ = \frac{9 \times 7^2 \times 5^2}{25 \times 7^2 \times 3^2} \]
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Cancel \(7^2\) from the numerator and denominator: \[ = \frac{9 \times 5^2}{25 \times 3^2} \]
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Recognize \(5^2 = 25\): \[ = \frac{9 \times 25}{25 \times 9} \]
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Cancel \(25\) and \(9\): \[ = 1 \]
Thus, the simplified value of the expression is \( \boxed{1} \).