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 simplify step by step.
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Simplifying the divisions:
- Recall that dividing by a fraction is equivalent to multiplying by its reciprocal: \[ a \div \left(\frac{b}{c}\right) = a \times \left(\frac{c}{b}\right) \] Applying this to our expression, we rewrite it 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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Next, we can combine like terms:
- Notice that there's a \(-3\) and a \(-\frac{5}{16}\) which can be simplified together once we distribute: \[ (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
Since \(\frac{7}{5}\) and \(-\frac{5}{16}\) will have a \(-5\) and a \(5\), we can simplify: \[ = (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
Which produces: \[ = (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(\frac{7}{16}\right) \]
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Cancelling Terms:
- Now let's group and cancel out some terms: \[ (-3) \times \left(\frac{4}{3}\right) = -4 \]
Then, substitute back: \[ -4 \times \left(\frac{2}{7}\right) \times \left(\frac{7}{16}\right) \]
In this grouping, notice the \( \frac{2}{7} \) and \( \frac{7}{16} \): \[ = -4 \times \left(\frac{2}{1}\right) \times \left(\frac{1}{16}\right) \]
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Final Calculation:
- Now calculate: \[ = -4 \times 2 \times \frac{1}{16} \] which is: \[ = -8 \times \frac{1}{16} = -\frac{8}{16} = -\frac{1}{2} \]
So the final result is: \[ \boxed{-\frac{1}{2}} \]
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