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Let's break down the mathematical expression step by step.

The expression consists of two parts:

\[
\left( \frac{3^{-1/6} x^{-1/3}}{y^{1/6} z^{1/3}} \right)^3 \times \left( \frac{x^{-1} z^{-2}}{2y^{-2}} \right)^{-1/2}
\]

### Step 1: Simplifying the first term
\[
\left( \frac{3^{-1/6} x^{-1/3}}{y^{1/6} z^{1/3}} \right)^3
\]
Distribute the power of 3 to each factor in the numerator and denominator:
\[
= \frac{(3^{-1/6})^3 \cdot (x^{-1/3})^3}{(y^{1/6})^3 \cdot (z^{1/3})^3}
\]
Simplify each exponent:
\[
= \frac{3^{-1/2} \cdot x^{-1}}{y^{1/2} \cdot z}
\]

### Step 2: Simplifying the second term
\[
\left( \frac{x^{-1} z^{-2}}{2y^{-2}} \right)^{-1/2}
\]
When raising to a power of \(-1/2\), we invert the fraction and raise each factor to the power \(1/2\):
\[
= \frac{(2y^{-2})^{1/2}}{(x^{-1} z^{-2})^{1/2}}
\]
Simplify each part:
\[
= \frac{\sqrt{2} \cdot y^{-1}}{x^{-1/2} \cdot z^{-1}}
\]
This becomes:
\[
= \frac{\sqrt{2} \cdot y^{-1}}{x^{-1/2} \cdot z^{-1}} = \sqrt{2} \cdot x^{1/2} \cdot z \cdot y^{-1}
\]

### Step 3: Multiplying the two terms
Now, multiply the two results:
\[
\left( \frac{3^{-1/2} \cdot x^{-1}}{y^{1/2} \cdot z} \right) \times \left( \sqrt{2} \cdot x^{1/2} \cdot z \cdot y^{-1} \right)
\]
Multiply the numerators and denominators separately:
\[
= \frac{3^{-1/2} \cdot \sqrt{2} \cdot x^{-1} \cdot x^{1/2} \cdot z \cdot z}{y^{1/2} \cdot y^{-1}}
\]
Simplify the exponents and coefficients:
\[
= \frac{\sqrt{2} \cdot 3^{-1/2} \cdot x^{-1/2} \cdot z^2}{y^{3/2}}
\]

Thus, the simplified result of the expression is:

\[
\frac{\sqrt{2} \cdot 3^{-1/2} \cdot x^{-1/2} \cdot z^2}{y^{3/2}}
\]

Would you like any further details or explanations?

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Here are 5 questions to expand on the information:

1. How do you simplify expressions with fractional exponents?
2. What are the rules for multiplying powers with the same base?
3. How does raising a fraction to a negative exponent affect the terms?
4. What is the significance of the square root in expressions like \(\sqrt{2}\)?
5. How do you deal with variables that have both positive and negative exponents?

**Tip**: When simplifying expressions with multiple variables and exponents, always focus on one operation at a time to avoid mistakes.

Simplify the mathematical expression involving fractional exponents.

Learn how to simplify an expression involving fractional exponents and powers. The problem includes raising fractions to powers and multiplying algebraic terms using exponentiation rules. Suitable for high school students in Grades 10-12.

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