To solve the expression \((-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \div \left(\frac{7}{15}\right)\), we will follow the order of operations and apply the properties of signed numbers.
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Rearranging and simplifying the division: Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as: \[ (-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \times \left(\frac{15}{7}\right) \]
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Grouping the factors: We can rearrange the order of multiplication: \[ (-4) \times \left(-\frac{3}{4}\right) \times \left(\frac{7}{5}\right) \times \left(\frac{15}{7}\right) \]
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Multiplying the signed numbers: First, we can compute \((-4) \times \left(-\frac{3}{4}\right)\): \[ (-4) \times \left(-\frac{3}{4}\right) = 4 \times \frac{3}{4} = 3 \]
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Now, substitute this back into the expression: Our expression now simplifies to: \[ 3 \times \left(\frac{7}{5}\right) \times \left(\frac{15}{7}\right) \]
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Simplifying further: Notice that \(\frac{7}{5} \times \frac{15}{7}\): \[ \frac{7}{5} \times \frac{15}{7} = \frac{15}{5} = 3 \]
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Final multiplication: Now, we multiply the result by 3: \[ 3 \times 3 = 9 \]
Thus, the final answer is: \[ \boxed{9} \]
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