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:

$\left(-\frac{7}{5} x y^{2} z\right) \times\left(\frac{13}{3} x^{2} y z^{2}\right)$

$=\left(-\frac{7}{5} \times \frac{13}{3}\right) \times\left(x \times x^{2}\right) \times\left(y^{2} \times y\right) \times\left(z \times z^{2}\right)$

$=\left(-\frac{7}{5} \times \frac{13}{3}\right) \times\left(x^{1+2}\right) \times\left(y^{2+1}\right) \times\left(z^{1+2}\right)$

$=-\frac{91}{15} x^{3} y^{3} x^{3}$

Thus, the answer is $-\frac{91}{15} x^{3} y^{3} x^{3}$.