Rewrite $\left(\frac{6}{7}\right)^{3}$ as $\frac{6}{7}\times \left(\frac{6}{7}\right)^{2}$. Cancel out $\left(\frac{6}{7}\right)^{2}$ in both numerator and denominator.
$$\frac{1}{\frac{6}{7}}$$
Divide $1$ by $\frac{6}{7}$ by multiplying $1$ by the reciprocal of $\frac{6}{7}$.
$$1\times \frac{7}{6}$$
Multiply $1$ and $\frac{7}{6}$ to get $\frac{7}{6}$.
