The problem asks you to simplify the expression $(5x^3y^4)^3$.

Step-by-Step Solution:

1. Expand the power: Use the power rule $(a^m)^n = a^{m \cdot n}$.

   • For $5$: $(5)^3 = 125$
   • For $x^3$: $(x^3)^3 = x^{3 \cdot 3} = x^9$
   • For $y^4$: $(y^4)^3 = y^{4 \cdot 3} = y^{12}$

2. Combine everything: Multiply the results: $(5x^3y^4)^3 = 125x^9y^{12}.$

Final Answer:

The simplified expression is: $125x^9y^{12}.$

Let me know if you’d like further explanation or details!

Here are five questions that expand on this topic:

1. What is the power rule, and how does it work with variables and constants?
2. How would the result change if the power were $(5x^3y^4)^2$ instead of $(5x^3y^4)^3$?
3. Can the same rules be applied to fractional or negative exponents?
4. What happens if there are additional variables or coefficients in the base expression?
5. How would you simplify an expression with nested exponents like $((x^2)^3)^4$?

Tip: Always simplify each part of the expression separately and apply exponent rules systematically!