Let's simplify the expressions one by one.

a. $(4x^3)^2$

Using the rule $(a^m)^n = a^{m \cdot n}$, we can simplify: $(4x^3)^2 = 4^2 \cdot (x^3)^2 = 16x^6$

b. $\left(\frac{-2}{x}\right)^3$

Here, apply the power rule to both the numerator and the denominator: $\left(\frac{-2}{x}\right)^3 = \frac{(-2)^3}{x^3} = \frac{-8}{x^3}$

c. $(5x^2y^4)^2$

Using the same power rule, apply the exponent to each term: $(5x^2y^4)^2 = 5^2 \cdot (x^2)^2 \cdot (y^4)^2 = 25x^4y^8$

Final Answers:

a. $16x^6$
b. $\frac{-8}{x^3}$
c. $25x^4y^8$

Would you like more details or explanations on any of these steps?

Here are 5 related questions for further practice:

1. Simplify $(3x^2y)^3$.
2. Simplify $\left(\frac{-3x^2}{2}\right)^2$.
3. What is the result of $(2x^5y^2)^3$?
4. Expand $\left(\frac{4x^3}{y}\right)^2$.
5. Simplify $(7x^2y^3)^4$.

Tip: When simplifying expressions with powers, make sure to apply the exponent to every term inside the parentheses.