https://file.aiquickdraw.com/mathbot/file/17274443880331e3ugxzl.jpg

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.

Simplify each expression: (5x^4)^2, (-3r^5)^3, (3a^3b^5)^4, (6m^{-4}n^7)^2

This guide simplifies exponential expressions like (5x^4)^2 and (6m^{-4}n^7)^2 using the power rule of exponents. Perfect for students in Grades 9-10, covering key algebraic concepts and exponent laws.

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