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 proceed step by step, using the properties of operations and simplifying where possible.
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Convert divisions to multiplications by taking the reciprocal:
\[ (-3) \div \left(\frac{3}{4}\right) = (-3) \times \left(\frac{4}{3}\right) \]
\[ \div \left(\frac{5}{7}\right) = \times \left(\frac{7}{5}\right) \]
Thus, the expression can be rewritten as:
\[ (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
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Combining all the fractions:
First, we can combine the multiplications:
\[ = (-3) \times \left(-\frac{5}{16}\right) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(\frac{7}{5}\right) \]
Notice that \( (-3) \times \left(-\frac{5}{16}\right) \) will yield a positive product:
\[ = \frac{15}{16} \]
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Now multiply by the remaining fractions:
The expression can thus be simplified to:
\[ = \frac{15}{16} \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(\frac{7}{5}\right) \]
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We can cancel terms before multiplying:
Notice that the \(\frac{7}{5}\) will cancel with \(-\frac{5}{16}\) as we already included it, and any other simplifications can be done now:
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Calculating the products step by step:
At this point, we combine the products:
\[ \frac{15 \times 4 \times 2 \times 7}{16 \times 3 \times 7 \times 5} \]
The \(7\)s cancel:
\[ = \frac{15 \times 4 \times 2}{16 \times 3 \times 5} \]
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Calculate numerators and denominators:
- Numerator: \(15 \times 4 \times 2 = 120\)
- Denominator: \(16 \times 3 \times 5 = 240\)
Thus, we have:
\[ = \frac{120}{240} \]
This simplifies to:
\[ = \frac{1}{2} \]
So the final answer is:
\[ \boxed{\frac{1}{2}} \]