https://file.aiquickdraw.com/mathbot/file/1730527039968vjv2q3gu.jpg

The expression in the image is:

\[
\left( \frac{x^2}{y^{-2}} \right)^{-3} \times \left( \frac{x^2}{y^3} \right)
\]

Let's simplify it step-by-step.

### Step 1: Simplify each term individually

1. **First Term**: \( \left( \frac{x^2}{y^{-2}} \right)^{-3} \)

   When raising a fraction to a power, each component of the fraction is raised to that power. Therefore:
   \[
   \left( \frac{x^2}{y^{-2}} \right)^{-3} = \frac{(x^2)^{-3}}{(y^{-2})^{-3}} = \frac{x^{-6}}{y^6} = \frac{1}{x^6 y^6}
   \]

2. **Second Term**: \( \frac{x^2}{y^3} \) (this one doesn't need any additional simplification)

### Step 2: Multiply the Simplified Terms

Now, we have:
\[
\frac{1}{x^6 y^6} \times \frac{x^2}{y^3} = \frac{x^2}{x^6 y^6 y^3}
\]

Combine like terms in the denominator:
- For \( x \): \( x^2 \) in the numerator and \( x^6 \) in the denominator combine as \( x^{2 - 6} = x^{-4} \).
- For \( y \): \( y^6 \) and \( y^3 \) in the denominator combine as \( y^{6 + 3} = y^9 \).

Thus, the expression becomes:
\[
\frac{x^2}{x^6 y^6 y^3} = \frac{1}{x^4 y^9} = x^{-4} y^{-9}
\]

### Final Answer

\[
\boxed{\frac{1}{x^4 y^9} \text{ or } x^{-4} y^{-9}}
\]

Would you like further details or have any questions about the steps?

---

### Related Questions

1. How would you simplify \(\left( \frac{x^3}{y^{-1}} \right)^{-2} \times \left( \frac{x^4}{y^2} \right)\)?
2. How does multiplying two expressions with exponents affect the exponents?
3. What happens when you raise a fraction with negative exponents to another negative exponent?
4. How can you rewrite an expression with negative exponents as a fraction with only positive exponents?
5. How would you handle an expression like \(\left( \frac{x^{-3}}{y^2} \right)^4\)?

---

### Tip

Remember, raising an expression with an exponent in the denominator to a power often reverses the sign of the exponent due to the reciprocal.

Simplify the expression \( \left( \frac{x^2}{y^{-2}} \right)^{-3} \times \left( \frac{x^2}{y^3} \right) \).

Step-by-step solution for simplifying the expression \( \left( \frac{x^2}{y^{-2}} \right)^{-3} \times \left( \frac{x^2}{y^3} \right) \). This problem covers algebraic concepts such as exponent rules and handling negative exponents, ideal for students in Grade 10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation