https://r2.erweima.ai/mathbot/image/1726457550222xlz1x1xc.jpg

The expression in the image appears to contain exponents and nested terms involving multiple variables. Let's break it down carefully and simplify it step by step.

### Expression Breakdown:
1. The first part is raised to the power of 3:
   \[
   \left( \frac{3^{-\frac{1}{6}} x^{-\frac{1}{3}}}{y^{\frac{1}{6}} z^{\frac{1}{3}}} \right)^3
   \]

2. The second part, multiplied with the first, is raised to the power of \(\frac{1}{2}\):
   \[
   \left( \frac{(x^{-1} z^{-2})}{2y^{-2}} \right)^{\frac{1}{2}}
   \]

### Step-by-Step Simplification:

#### 1. Simplifying the first part:
   \[
   \left( \frac{3^{-\frac{1}{6}} x^{-\frac{1}{3}}}{y^{\frac{1}{6}} z^{\frac{1}{3}}} \right)^3
   \]

   Apply the power of 3 to each component:
   \[
   \frac{(3^{-\frac{1}{6}})^3 (x^{-\frac{1}{3}})^3}{(y^{\frac{1}{6}})^3 (z^{\frac{1}{3}})^3}
   \]
   Which simplifies to:
   \[
   \frac{3^{-\frac{1}{2}} x^{-1}}{y^{\frac{1}{2}} z}
   \]

#### 2. Simplifying the second part:
   \[
   \left( \frac{x^{-1} z^{-2}}{2y^{-2}} \right)^{\frac{1}{2}}
   \]
   Apply the power of \(\frac{1}{2}\) to each component:
   \[
   \frac{(x^{-1})^{\frac{1}{2}} (z^{-2})^{\frac{1}{2}}}{(2)^{\frac{1}{2}} (y^{-2})^{\frac{1}{2}}}
   \]
   Simplifies to:
   \[
   \frac{x^{-\frac{1}{2}} z^{-1}}{\sqrt{2} y^{-1}}
   \]
   Which becomes:
   \[
   \frac{x^{-\frac{1}{2}} z^{-1} y}{\sqrt{2}}
   \]

#### 3. Combine both parts:
   Now, multiply both simplified parts:
   \[
   \frac{3^{-\frac{1}{2}} x^{-1} z}{y^{\frac{1}{2}}} \times \frac{x^{-\frac{1}{2}} z^{-1} y}{\sqrt{2}}
   \]
   Combine the terms:
   \[
   \frac{3^{-\frac{1}{2}} x^{-1} x^{-\frac{1}{2}} y z z^{-1}}{y^{\frac{1}{2}} \sqrt{2}}
   \]
   Simplify the powers of \(x\) and \(z\):
   \[
   \frac{3^{-\frac{1}{2}} x^{-\frac{3}{2}} y^{\frac{1}{2}}}{\sqrt{2}}
   \]

### Final Simplified Expression:
\[
\frac{3^{-\frac{1}{2}} x^{-\frac{3}{2}} y^{\frac{1}{2}}}{\sqrt{2}}
\]

This is the fully simplified form of the given expression.

Would you like more details or have any further questions?

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. What are the rules for combining exponents with different bases?
3. How does the distributive property work when dealing with exponents?
4. Can fractional exponents be rewritten as roots?
5. What is the significance of negative exponents in division?

### Tip:
Always apply exponent rules (product, quotient, and power of a power) carefully when simplifying complex expressions!

Simplify the given algebraic expression with multiple variables and exponents.

Learn how to simplify a complex algebraic expression involving fractional and negative exponents across multiple variables. This problem covers key algebraic concepts such as exponent rules and simplification techniques. Suitable for high school students in Grades 9-12.

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