https://file.aiquickdraw.com/mathbot/file/17257586893208x19szn7.jpg

The image contains a math problem involving exponents and radical expressions. I will analyze the image to extract the mathematical content and provide a solution.

### Extracted Problem

The problem appears to involve rewriting an expression with a negative exponent. The expression given in the image is:

\[
\left( \frac{x^4}{y^5} \right)^{-3}
\]

### Solution

To simplify the expression with a negative exponent, we can use the rule:

\[
a^{-n} = \frac{1}{a^n}
\]

Applying this rule to the entire expression:

\[
\left( \frac{x^4}{y^5} \right)^{-3} = \frac{1}{\left( \frac{x^4}{y^5} \right)^3}
\]

Now, apply the exponent to both the numerator and the denominator:

\[
\frac{1}{\frac{x^{4 \cdot 3}}{y^{5 \cdot 3}}} = \frac{1}{\frac{x^{12}}{y^{15}}}
\]

To simplify further, take the reciprocal:

\[
\frac{y^{15}}{x^{12}}
\]

### Final Answer

\[
\boxed{\frac{y^{15}}{x^{12}}}
\]

Would you like more details, or do you have any questions?

### Related Questions

1. What are the rules for simplifying expressions with exponents?
2. How do you simplify expressions with fractional exponents?
3. How does the power of a quotient rule work with different bases?
4. What is the importance of understanding negative exponents?
5. How do you convert expressions with exponents to radical form?

### Tip

Remember, when dealing with negative exponents, the base and its exponent simply switch to their reciprocal position and the exponent becomes positive.

Simplify the expression: \(\left( \frac{x^4}{y^5} \right)^{-3}\)

Learn how to simplify expressions involving negative exponents like \(\left( \frac{x^4}{y^5} \right)^{-3}\). This step-by-step solution includes key algebraic concepts and theorems such as the Negative Exponent Rule and Power of a Quotient Rule. Ideal for students in Grades 9-12.

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