The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ because $\frac{2}{3} \times \frac{3}{2} = 1$. Dividing by a fraction equals multiplying by its reciprocal: $8 \div \frac{2}{3} = 8 \times \frac{3}{2}$. In general, for any nonzero fraction $\frac{b}{c}$, $a \div \frac{b}{c} = a \times \frac{c}{b}$.

The reciprocal of $-\frac{3}{4}$ is $-\frac{4}{3}$ because $\left(-\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = 1$. So $-6 \div \left(-\frac{3}{4}\right) = -6 \times \left(-\frac{4}{3}\right)$. This follows the rule $a \div \frac{b}{c} = a \times \frac{c}{b}$.

The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$ because $\frac{5}{7} \times \frac{7}{5} = 1$. The opposite (negative) changes the sign, not the numerator and denominator, so it is not the reciprocal.

Dividing by a fraction equals multiplying by its reciprocal: $a \div \frac{b}{c} = a \times \frac{c}{b}$. So $12 \div \frac{3}{5} = 12 \times \frac{5}{3}$. Both sides evaluate to $20$.

The reciprocal of $\frac{7}{10}$ is $\frac{10}{7}$ because $\frac{7}{10} \times \frac{10}{7} = 1$. So $7 \div \frac{7}{10} = 7 \times \frac{10}{7} = 10$. Multiplying by the original fraction would give $7 \times \frac{7}{10} = \frac{49}{10}$, which is not correct for division.