https://file.aiquickdraw.com/mathbot/file/1728539558279bkakcune.jpg

The expression in the image is:

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

Now, let's simplify the expression step by step:

### Step 1: Apply exponent rules inside both parentheses
For the numerator:
\[
(x^3 \cdot y^{-2} \cdot z^4)^{-2} = x^{3 \cdot (-2)} \cdot y^{-2 \cdot (-2)} \cdot z^{4 \cdot (-2)}
= x^{-6} \cdot y^{4} \cdot z^{-8}
\]

For the denominator:
\[
(x^{-1} \cdot y^4 \cdot z^{-2})^3 = x^{-1 \cdot 3} \cdot y^{4 \cdot 3} \cdot z^{-2 \cdot 3}
= x^{-3} \cdot y^{12} \cdot z^{-6}
\]

### Step 2: Combine the two fractions
Now, we have:
\[
\frac{x^{-6} \cdot y^{4} \cdot z^{-8}}{x^{-3} \cdot y^{12} \cdot z^{-6}}
\]

### Step 3: Use the property of division of exponents
When dividing powers with the same base, subtract the exponents:
\[
x^{-6 - (-3)} = x^{-6 + 3} = x^{-3}
\]
\[
y^{4 - 12} = y^{-8}
\]
\[
z^{-8 - (-6)} = z^{-8 + 6} = z^{-2}
\]

### Final Simplified Expression:
\[
\boxed{x^{-3} \cdot y^{-8} \cdot z^{-2}}
\]

Or equivalently:
\[
\frac{1}{x^3 \cdot y^8 \cdot z^2}
\]

Would you like further details or clarifications? Here are some related questions you might consider:

1. How do exponent rules work in general for multiplication and division?
2. What are the properties of negative exponents?
3. How does fractional exponentiation impact the simplification process?
4. How can I express the result using only positive exponents?
5. What is the difference between simplifying an expression and solving an equation?

**Tip:** When simplifying exponents, always keep track of negative signs carefully to avoid mistakes!

Simplify the expression: \frac{(x^3 \cdot y^{-2} \cdot z^4)^{-2}}{(x^{-1} \cdot y^4 \cdot z^{-2})^3}

Learn how to simplify the complex exponent expression \frac{(x^3 \cdot y^{-2} \cdot z^4)^{-2}}{(x^{-1} \cdot y^4 \cdot z^{-2})^3} using exponent rules. This problem covers key algebraic concepts such as the power of a power rule, multiplication and division of powers, and handling negative exponents. Suitable for students in Grades 9-11.

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