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:

$\left(\frac{4}{3} u^{2} v w\right) \times\left(-5 u v w^{2}\right) \times\left(\frac{1}{3} v^{2} w u\right)$

$=\left\{\frac{4}{3} \times(-5) \times \frac{1}{3}\right\} \times\left(u^{2} \times u \times u\right) \times\left(v \times v \times v^{2}\right) \times\left(w \times w^{2} \times w\right)$

$=\left\{\frac{4}{3} \times(-5) \times \frac{1}{3}\right\} \times\left(u^{2+1+1}\right) \times\left(v^{1+1+2}\right) \times\left(w^{1+2+1}\right)$

$=-\frac{20}{9} u^{4} v^{4} w^{4}$

Thus, the answer is $-\frac{20}{9} u^{4} v^{4} w^{4}$.