https://file.aiquickdraw.com/mathbot/file/1726497880239xnsqk70w.jpg

The expression in the image is:

\[
\left( \frac{64x^2}{49y^4} \right)^{-\frac{3}{2}}
\]

To simplify this expression, we will apply exponent rules:

### Step-by-step simplification:

1. **Apply the power to both numerator and denominator:**
   \[
   \left( \frac{64x^2}{49y^4} \right)^{-\frac{3}{2}} = \frac{(64x^2)^{-\frac{3}{2}}}{(49y^4)^{-\frac{3}{2}}}
   \]

2. **Simplify the numerator:**
   \[
   (64x^2)^{-\frac{3}{2}} = 64^{-\frac{3}{2}} \cdot (x^2)^{-\frac{3}{2}} = (64^{\frac{1}{2}})^{-3} \cdot x^{-3} = 8^{-3} \cdot x^{-3} = \frac{1}{512x^3}
   \]

3. **Simplify the denominator:**
   \[
   (49y^4)^{-\frac{3}{2}} = 49^{-\frac{3}{2}} \cdot (y^4)^{-\frac{3}{2}} = (49^{\frac{1}{2}})^{-3} \cdot y^{-6} = 7^{-3} \cdot y^{-6} = \frac{1}{343y^6}
   \]

4. **Combine numerator and denominator:**
   \[
   \frac{1}{512x^3} \div \frac{1}{343y^6} = \frac{343y^6}{512x^3}
   \]

Thus, the simplified expression is:

\[
\frac{343y^6}{512x^3}
\]

Would you like further details or have any questions?

### Related questions:
1. How can you apply negative exponents to more complex algebraic expressions?
2. What are the rules for simplifying expressions with fractional exponents?
3. How do you handle variables with exponents during simplification?
4. Can you explain the concept of raising a fraction to a negative exponent in general terms?
5. How does the simplification change if we modify the powers of \(x\) or \(y\)?

**Tip:** When simplifying expressions with exponents, always break them down step-by-step, focusing on applying exponent rules to both the numerator and the denominator.

Learn how to simplify the expression (64x^2 / 49y^4)^(-3/2) step-by-step using exponent rules. This problem involves applying negative exponents, fractional exponents, and the laws of exponents to simplify algebraic fractions. Suitable for students in Grades 10-12.

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