The given mathematical expression can be simplified as follows:

$a^2 \sqrt{b} \log\left(a^{\frac{2}{3}}\right) \log\left(\frac{1}{b^4}\right) \log\left(c^6\right) \frac{1}{c}$

This expands into:

$\frac{a^2 \sqrt{b} \log\left(a^{\frac{2}{3}}\right) \log\left(\frac{1}{b^4}\right) \log\left(c^6\right)}{c \log(10)^3}$

Would you like me to proceed with further simplifications or solve it under specific conditions (like setting values for $a$, $b$, and $c$)?

Here are 5 related questions you might explore:

1. How can we simplify logarithmic expressions involving products and powers?
2. What properties of logarithms are useful in solving this equation?
3. How does changing the base of a logarithm affect the result?
4. How does the term involving $\sqrt{b}$ affect the simplification?
5. What happens if specific values for $a$, $b$, and $c$ are given?

Tip: The property $\log(a^b) = b \log(a)$ is often helpful when simplifying logarithmic expressions with exponents.