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Let's simplify the given expression:

\[
\left( -2x^3y^4 \right)^5
\]

### Step-by-step simplification:

1. Apply the power of 5 to each factor inside the parentheses. This means raising \(-2\), \(x^3\), and \(y^4\) all to the power of 5:
   \[
   (-2)^5 \cdot (x^3)^5 \cdot (y^4)^5
   \]

2. Simplify each term:
   - \((-2)^5 = -32\)
   - \((x^3)^5 = x^{3 \cdot 5} = x^{15}\)
   - \((y^4)^5 = y^{4 \cdot 5} = y^{20}\)

3. Combine the simplified terms:
   \[
   -32x^{15}y^{20}
   \]

Thus, the simplified expression is:

\[
-32x^{15}y^{20}
\]

Let me know if you need further details, or have questions!

### Related Questions:
1. How would the simplification change if the exponent outside the parentheses were different, say \(6\)?
2. Can you explain how the properties of exponents work for negative bases?
3. What if the original expression had been \( (2x^3y^4)^5 \) without the negative sign?
4. How can we generalize this approach to simplify expressions like \( (ax^m y^n)^p \)?
5. What happens if you apply fractional exponents instead of integer exponents?

### Tip:
When simplifying expressions with exponents, always distribute the outer exponent to each factor inside the parentheses before simplifying individual terms!

Simplify (-2x^3y^4)^5 and write your answer without parentheses.

Learn how to simplify the expression (-2x^3y^4)^5 using exponent rules. This problem involves applying the power of a power rule and simplifying polynomials, suitable for students in Grades 9-11.

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