https://file.aiquickdraw.com/mathbot/file/1725575284984899m7n5a.jpg

The expression in the image can be simplified as follows:

1. **Understanding the expression**: The given expression is
   \[
   \left(\frac{-2x^{0}}{2^{-3}z}\right)^{-2} \cdot x^{-3}y^{4}z^{2}.
   \]
   
2. **Simplifying the fraction inside the parentheses**: 
   - Since \(x^0 = 1\), the numerator simplifies to \(-1\).
   - The denominator \(2^{-3}\) simplifies to \(\frac{1}{2^3} = \frac{1}{8}\).
   - Therefore, the entire fraction becomes \(\frac{-1}{\frac{1}{8}z} = -8z\).

3. **Applying the power of \(-2\) to the fraction**:
   \[
   \left(-8z\right)^{-2} = \left((-8)^{-2}\right)(z^{-2}) = \frac{1}{64}z^{-2}.
   \]

4. **Combining with the rest of the terms**:
   \[
   \frac{1}{64}z^{-2} \cdot x^{-3}y^{4}z^{2}.
   \]

5. **Final simplification by combining like terms**:
   - The \(z^{-2}\) and \(z^2\) cancel each other out.
   - This leaves:
   \[
   \frac{1}{64}x^{-3}y^{4}.
   \]
   - Alternatively, in fractional form:
   \[
   \frac{y^{4}}{64x^{3}}.
   \]

So, the simplified expression is \(\frac{y^{4}}{64x^{3}}\).

Would you like any further details or have any questions about this? Here are some related queries you might consider:

1. How does exponentiation work with negative and fractional bases?
2. Can you explain the properties used to simplify the powers of variables?
3. What are some common mistakes to avoid when simplifying complex algebraic expressions?
4. How does simplification differ when variables are under a radical?
5. Why do the \(z^{-2}\) and \(z^2\) terms cancel out?

**Tip**: Always ensure that any intermediate negative exponents are correctly simplified before combining like terms. This helps avoid errors in the final result.

Simplify (-2x^0 / y^2z^-3)^-2 * x^-3 y^4 z^2

Simplify the expression (-2x^0 / y^2z^-3)^-2 * x^-3 y^4 z^2 using exponent rules. This example covers negative and zero exponents and applies exponent multiplication and distribution rules. Suitable for high school algebra students.

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