Find each of the following product:

To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$.

We have:

$\left(\frac{7}{9} a b^{2}\right) \times\left(\frac{15}{7} a c^{2} b\right) \times\left(-\frac{3}{5} a^{2} c\right)$

$=\left\{\frac{7}{9} \times \frac{15}{7} \times\left(-\frac{3}{5}\right)\right\} \times\left(a \times a \times a^{2}\right) \times\left(b^{2} \times b\right) \times\left(c^{2} \times c\right)$

$=-a^{4} b^{3} c^{3}$

Thus, the answer is $-a^{4} b^{3} c^{3}$.