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Bot GPT 4-o mini To solve the expression \((-3) \div \left(\frac{3}{4}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \div \left(\frac{5}{7}\right)\), let's break it down step by step.
**Step 1: Rewrite the divisions as multiplications by the reciprocal.**
\[
(-3) \div \left(\frac{3}{4}\right) = (-3) \times \left(\frac{4}{3}\right)
\]
\[
-\frac{5}{16} \div \left(\frac{5}{7}\right) = -\frac{5}{16} \times \left(\frac{7}{5}\right)
\]
Now the expression can be rewritten as:
\[
(-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right)
\]
**Step 2: Simplify the terms.**
\[
(-3) \times \left(\frac{4}{3}\right) = -\frac{12}{3} = -4
\]
Now the expression simplifies to:
\[
-4 \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right)
\]
**Step 3: Simplify further.**
Multiplying \(-4\) and \(-\frac{5}{16}\):
\[
-4 \times -\frac{5}{16} = \frac{20}{16} = \frac{5}{4}
\]
Now we have:
\[
\frac{5}{4} \times \left(\frac{2}{7}\right) \times \left(\frac{7}{5}\right)
\]
**Step 4: Multiply the fractions.**
Notice that \(\frac{7}{5}\) cancels with \(\frac{5}{4}\) when we multiply:
\[
\frac{5}{4} \times \frac{2}{7} \times \frac{7}{5} = \frac{5 \cdot 2 \cdot 7}{4 \cdot 7 \cdot 5}
\]
The \(5\) cancels out:
\[
= \frac{2}{4} = \frac{1}{2}
\]
Thus, the final answer is:
\[
\boxed{\frac{1}{2}}
\]