Find $\left(\frac{5}{7}\right)^{3}\left(\frac{4}{5}\right)^{3}$.
Recall that if $b$ is nonzero and $n$ is a positive integer, then $$\left(\frac{a}{b}\right)^{n}=\frac{a^n}{b^n}.$$For this problem we have $$\left(\frac{5}{7}\right)^{3}\left(\frac{4}{5}\right)^{3}=\frac{5^3}{7^3}\cdot \frac{4^3}{5^3}=\frac{5^3\cdot 4^3}{7^3\cdot 5^3}.$$We can simplify because there is a factor of $5^3$ in the numerator and the denominator.  To do this we use the commutative property of multiplication and use the multiplication rule in reverse as follows $$\frac{5^3\cdot 4^3}{7^3\cdot 5^3}=\frac{4^3\cdot 5^3}{7^3\cdot 5^3}=\frac{4^3}{7^3}\cdot \frac{5^3}{5^3}=\frac{4^3}{7^3}\cdot 1=\frac{4^3}{7^3}=\boxed{\frac{64}{343}}.$$We also might have noted that \[\left(\frac{5}{7}\right)^{3}\left(\frac{4}{5}\right)^{3}=\left(\frac57\cdot \frac45\right)^3 = \left(\frac47\right)^3 = \frac{4^3}{7^3} = \boxed{\frac{64}{343}}.\]