https://r2.erweima.ai/mathbot/image/1726456256521x1m1e3ke.jpg

The expression in the image appears to be:

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

I'll simplify this step by step:

### Step 1: Simplify the first part of the expression

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

This means raising both the numerator and denominator to the power of 3:

\[
\frac{(3^{-\frac{1}{6}})^3 (x^{-\frac{1}{3}})^3}{(y^{\frac{1}{6}})^3 (z^{\frac{1}{3}})^3}
\]

This simplifies to:

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

### Step 2: Simplify the second part of the expression

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

Raising each part of the fraction to the power of \(\frac{1}{2}\):

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

This simplifies to:

\[
\frac{x^{-\frac{1}{2}} z^{-\frac{1}{4}}}{\sqrt{2} y^{-1}}
\]

### Step 3: Multiply the two parts together

Now, multiply the two simplified expressions:

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

This becomes:

\[
\frac{3^{-\frac{1}{2}} x^{-1} x^{-\frac{1}{2}} z^{-1} z^{-\frac{1}{4}} y^1}{\sqrt{2} y^{\frac{1}{2}}}
\]

### Step 4: Combine like terms

For \(x\):

\[
x^{-1} x^{-\frac{1}{2}} = x^{-\frac{3}{2}}
\]

For \(z\):

\[
z^{-1} z^{-\frac{1}{4}} = z^{-\frac{5}{4}}
\]

For \(y\):

\[
y^1 y^{-\frac{1}{2}} = y^{\frac{1}{2}}
\]

Thus, the final simplified expression is:

\[
\frac{3^{-\frac{1}{2}} x^{-\frac{3}{2}} z^{-\frac{5}{4}} y^{\frac{1}{2}}}{\sqrt{2}}
\]

Let me know if you'd like further details or clarifications!

---

Here are 5 related questions:
1. How would the result change if a different exponent were applied to one of the variables?
2. Can you express the result in terms of logarithms?
3. How does the power of a fraction affect both the numerator and the denominator?
4. What are the general rules for simplifying products of powers with different exponents?
5. How does introducing negative exponents affect the final expression?

**Tip**: When simplifying powers, always treat each term separately, then combine like terms to avoid mistakes!

Simplify the following expression: ((3^(-1/6) * x^(-1/3)) / (y^(1/6) * z^(1/3)))^3 * ((x^(-1) * z^(-1/2)) / (2 * y^(-2)))^(1/2)

Learn how to simplify a complex algebraic expression involving negative and fractional exponents. This step-by-step solution uses key exponent rules to simplify the given expression. Suitable for high school students in Grades 10-12.

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