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:

$\left(-\frac{24}{25} x^{3} z\right) \times\left(-\frac{15}{16} x z^{2} y\right)$

$=\left\{\left(-\frac{24}{25}\right) \times\left(-\frac{15}{16}\right)\right\} \times\left(x^{3} \times x\right) \times\left(z \times z^{2}\right) \times y$

$=\left\{\left(-\frac{24}{25}\right) \times\left(-\frac{15}{16}\right)\right\} \times\left(x^{3+1}\right) \times\left(z^{1+2}\right) \times y$

$=\frac{9}{10} x^{4} y z^{3}$

Thus, the answer is $\frac{9}{10} x^{4} y z^{3}$.