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 keep track of the signs.
Step 1: Rewrite the division as multiplication by the reciprocal
First, we rewrite the division \(\div \left(\frac{7}{15}\right)\) as multiplication by the reciprocal:
\[ (-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \times \left(\frac{15}{7}\right) \]
Step 2: Group the fractions
Now, we can group the multiplication:
\[ (-4) \times \left(-\frac{3}{4}\right) \times \left(\frac{7}{5}\right) \times \left(\frac{15}{7}\right) \]
We can observe that \(\frac{7}{5} \times \frac{15}{7}\) can simplify nicely, as the 7s will cancel out.
Step 3: Simplify \( \frac{7}{5} \times \frac{15}{7} \)
\[ \frac{7}{5} \times \frac{15}{7} = \frac{7 \times 15}{5 \times 7} = \frac{15}{5} = 3 \]
Step 4: Substitute back into the expression
Now substituting this back into the original expression, we have:
\[ (-4) \times \left(-\frac{3}{4}\right) \times 3 \]
Step 5: Calculate \((-4) \times \left(-\frac{3}{4}\right)\)
Multiplying the first two terms:
\[ (-4) \times \left(-\frac{3}{4}\right) = 4 \times \frac{3}{4} = 3 \]
(Note: The two negatives make a positive.)
Step 6: Now multiply by the remaining 3
Now we multiply the result by 3:
\[ 3 \times 3 = 9 \]
Conclusion
Putting it all together, the value of the original expression \((-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \div \left(\frac{7}{15}\right)\) is:
\[ \boxed{9} \]
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