https://r2.erweima.ai/mathbot/image/17264333093440ae7s6wg.jpg

The expression provided in the image is:

\[
\left( \frac{(2x)^2 y^{-1}}{z^{-2}} \right)^3 (xyz)^{-2} \left( \frac{z \cdot x}{z^4} \right)^{-3}
\]

I will simplify this step by step:

### Step 1: Simplify the first part
\[
\left( \frac{(2x)^2 y^{-1}}{z^{-2}} \right)^3
\]

Start by simplifying the inner expression:
- \((2x)^2 = 4x^2\),
- \(y^{-1}\) remains as is,
- \(z^{-2}\) is in the denominator, so it becomes \(z^2\) in the numerator.

Thus, the expression becomes:
\[
\left( \frac{4x^2 y^{-1} z^2}{1} \right)^3 = (4x^2 y^{-1} z^2)^3
\]

Now, apply the cube to each term:
\[
4^3 \cdot (x^2)^3 \cdot (y^{-1})^3 \cdot (z^2)^3 = 64x^6 y^{-3} z^6
\]

### Step 2: Simplify the second part
\[
(xyz)^{-2}
\]

Apply the power of \(-2\) to each term:
\[
(xyz)^{-2} = x^{-2} y^{-2} z^{-2}
\]

### Step 3: Simplify the third part
\[
\left( \frac{z \cdot x}{z^4} \right)^{-3}
\]

Simplify the fraction first:
\[
\frac{z \cdot x}{z^4} = \frac{x}{z^3}
\]

Now, apply the power of \(-3\):
\[
\left( \frac{x}{z^3} \right)^{-3} = \frac{z^9}{x^3}
\]

### Step 4: Combine all parts
Now, combine the three simplified expressions:
\[
64x^6 y^{-3} z^6 \cdot x^{-2} y^{-2} z^{-2} \cdot \frac{z^9}{x^3}
\]

### Step 5: Simplify the final expression
Combine powers of like terms:
- For \(x\): \(x^6 \cdot x^{-2} \cdot x^{-3} = x^{6 - 2 - 3} = x^1\),
- For \(y\): \(y^{-3} \cdot y^{-2} = y^{-5}\),
- For \(z\): \(z^6 \cdot z^{-2} \cdot z^9 = z^{6 - 2 + 9} = z^{13}\).

Thus, the final simplified expression is:
\[
64x y^{-5} z^{13}
\]

Let me know if you have any questions or need further clarification!

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Here are 5 related questions you might find interesting:
1. What rules govern the simplification of negative exponents?
2. How does exponentiation distribute over multiplication and division?
3. What are some techniques to handle complex expressions with multiple variables?
4. How would the result change if the powers in the expression were fractional rather than integers?
5. How does the order of operations affect the simplification process in this expression?

**Tip**: Always group similar terms and handle exponents before multiplication to avoid errors in simplification!

Simplify the following expression: ((2x)^2 / Z^-2 * y^-1)^3 * (x * y * z)^-2 * ((z * x) / z^4)^-3.

Learn how to simplify complex algebraic expressions with exponents. This step-by-step guide covers important concepts like negative exponents, power rules, and multiplying terms with exponents. Suitable for students in Grades 9-12.

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