To solve the expression \( (-3) \div \left( \frac{3}{4} \right) \times \left( \frac{2}{7} \right) \times \left( -\frac{5}{16} \right) \div \left( \frac{5}{7} \right) \), we will use the properties of operations step by step.
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Rewrite the division as multiplication by the reciprocal: \[ (-3) \div \left( \frac{3}{4} \right) = (-3) \times \left( \frac{4}{3} \right) \] Then our expression becomes: \[ (-3) \times \left( \frac{4}{3} \right) \times \left( \frac{2}{7} \right) \times \left( -\frac{5}{16} \right) \div \left( \frac{5}{7} \right) \] Now we can also rewrite the division: \[ \div \left( \frac{5}{7} \right) = \times \left( \frac{7}{5} \right) \] Our expression now looks like: \[ (-3) \times \left( \frac{4}{3} \right) \times \left( \frac{2}{7} \right) \times \left( -\frac{5}{16} \right) \times \left( \frac{7}{5} \right) \]
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Now multiply all parts together: \[ = (-3) \times \left( \frac{4}{3} \right) \times \left( \frac{2}{7} \right) \times \left( -\frac{5}{16} \right) \times \left( \frac{7}{5} \right) \]
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Group the factors: \[ = (-3) \times \left( -\frac{5}{16} \right) \times \left( \frac{4}{3} \right) \times \left( \frac{2}{7} \right) \times \left( \frac{7}{5} \right) \]
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Cancel out terms where possible: \[ (-3) \times \left( -\frac{5}{16} \right) \times \left( \frac{4}{3} \right) \times \left( \frac{2}{7} \right) \times \left( \frac{7}{5} \right) = \frac{(-3) \times (-5) \times 4 \times 2 \times 7}{16 \times 3 \times 7 \times 5} \] Notice that \(5\) in the numerator cancels with \(5\) in the denominator and \(7\) cancels as well: \[ = \frac{(-3) \times (-5) \times 4 \times 2}{16 \times 3} \]
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Calculate the numerator and denominator:
- Numerator: \[ (-3) \times (-5) = 15 \] \[ 15 \times 4 = 60 \] \[ 60 \times 2 = 120 \]
- Denominator: \[ 16 \times 3 = 48 \]
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Now, we have: \[ \frac{120}{48} \]
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Simplify the fraction: \[ \frac{120 \div 24}{48 \div 24} = \frac{5}{2} \]
Thus, the final answer is: \[ \boxed{\frac{5}{2}} \]