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Let's analyze the given expression and find its equivalent.

The expression inside the parentheses is:

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

To solve this, we first simplify the expression inside the parentheses by eliminating the negative exponent. Here's the step-by-step process:

1. **Apply the exponent**: When you raise a fraction to a negative power, you can reciprocate the fraction and change the sign of the exponent. So the expression becomes:

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

2. **Expand the exponent**: Now, distribute the exponent \(-2\) to each part of the product inside:

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

3. **Simplify the exponents**:

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

4. **Rewrite the expression**: Combine the factors into a single fraction:

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

This matches the third answer choice in your options.

So, the correct equivalent expression is:

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

Would you like a detailed explanation of any step? 

Here are 5 related questions to expand your understanding:

1. How do you simplify expressions with negative exponents?
2. What happens when you raise a fraction to a negative power?
3. How do you distribute an exponent over a product of variables?
4. What is the general rule for multiplying exponents?
5. How do you handle fractional exponents?

**Tip:** When dealing with complex expressions, breaking them down into smaller steps can help make the process easier to manage.

Which expression is equivalent to \( \left( \frac{1}{x^{1/2} y^{-1/4} z} \right)^{-2} \)?

Discover how to simplify the expression \( \left( \frac{1}{x^{1/2} y^{-1/4} z} \right)^{-2} \) using key algebraic concepts like negative and fractional exponents. This problem is suitable for students in Grades 9-10.

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