Here are the simplified forms of the mathematical expressions displayed on the uploaded image:

A. $(5x^2)^3$

• Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$
• $(5x^2)^3 = 5^3 \cdot (x^2)^3 = 125x^6$

B. $(-3r^5)^3$

• Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$
• $(-3r^5)^3 = (-3)^3 \cdot (r^5)^3 = -27r^{15}$

D. $(3a^3b^5)^4$

• Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$
• $(3a^3b^5)^4 = 3^4 \cdot (a^3)^4 \cdot (b^5)^4 = 81a^{12}b^{20}$

E. $(6m^{-4}n^7)^2$

• Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$
• $(6m^{-4}n^7)^2 = 6^2 \cdot (m^{-4})^2 \cdot (n^7)^2 = 36m^{-8}n^{14}$

Would you like further details or explanations on any of these solutions?

Further questions you might find interesting:

1. What would be the simplified form of $(2x^2y^3)^5$?
2. How would you simplify $(4a^3)^3 \cdot (2a^2)^2$?
3. If a term like $(x^{-2}y^4)^3$ appears, what steps should you follow to simplify it?
4. Can you explain how to simplify the expression $(7m^2n^{-3})^4$?
5. What would the result be if you simplify $(2b^2c^{-3})^5$?

Tip: Always double-check your calculations by plugging in small values for the variables to confirm that the simplified form behaves as expected mathematically.