Which Expression Is Equivalent To $\left(x^{\frac{1}{2}} Y^{-\frac{1}{4}} Z\right)^{-2}$?A. $\frac{x^{\frac{1}{2}}}{y Z^2}$B. $\frac{x^{\frac{1}{2}}}{y^{\frac{1}{4}} Z^2}$C. $\frac{y^{\frac{1}{2}}}{x Z^2}$

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying the expression $\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2}$ and explore the different options available.

Understanding Exponents

Before we dive into the problem, let's review the basics of exponents. An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $x^2$, the exponent 2 represents the power to which the base $x$ is raised.

Simplifying the Expression

Now, let's simplify the expression $\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2}$. To do this, we need to apply the rules of exponents. When we raise a power to another power, we multiply the exponents. In this case, we have:

$$\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2} = x^{\frac{1}{2} \cdot (-2)} y^{-\frac{1}{4} \cdot (-2)} z^{-2}$$

Applying the Rules of Exponents

Now, let's apply the rules of exponents to simplify the expression further. When we multiply two powers with the same base, we add the exponents. In this case, we have:

$$x^{\frac{1}{2} \cdot (-2)} y^{-\frac{1}{4} \cdot (-2)} z^{-2} = x^{-1} y^{\frac{1}{2}} z^{-2}$$

Rewriting the Expression

Now, let's rewrite the expression in a more familiar form. We can rewrite $x^{-1}$ as $\frac{1}{x}$ and $z^{-2}$ as $\frac{1}{z^2}$. In this case, we have:

$$x^{-1} y^{\frac{1}{2}} z^{-2} = \frac{1}{x} y^{\frac{1}{2}} \frac{1}{z^2}$$

Simplifying the Expression Further

Now, let's simplify the expression further. We can rewrite $y^{\frac{1}{2}}$ as $\sqrt{y}$. In this case, we have:

$$\frac{1}{x} y^{\frac{1}{2}} \frac{1}{z^2} = \frac{1}{x} \sqrt{y} \frac{1}{z^2}$$

Evaluating the Options

Now, let's evaluate the options available. We have three options:

A. $\frac{x^{\frac{1}{2}}}{y z^2}$ B. $\frac{x^{\frac{1}{2}}}{y^{\frac{1}{4}} z^2}$ C. $\frac{y^{\frac{1}{2}}}{x z^2}$

Option A

Let's evaluate option A. We have:

$$\frac{x^{\frac{1}{2}}}{y z^2} = \frac{x^{\frac{1}{2}}}{y} \frac{1}{z^2}$$

This is not equal to the simplified expression.

Option B

Let's evaluate option B. We have:

$$\frac{x^{\frac{1}{2}}}{y^{\frac{1}{4}} z^2} = \frac{x^{\frac{1}{2}}}{\sqrt{y}} \frac{1}{z^2}$$

This is not equal to the simplified expression.

Option C

Let's evaluate option C. We have:

$$\frac{y^{\frac{1}{2}}}{x z^2} = \frac{\sqrt{y}}{x} \frac{1}{z^2}$$

This is equal to the simplified expression.

Conclusion

In this article, we simplified the expression $\left(x^{\frac{1}{2}} y^{-\frac{1}{4}} z\right)^{-2}$ and evaluated the different options available. We found that option C is the correct answer.

Final Answer

The final answer is $\boxed{C}$.

Additional Resources

For more information on simplifying exponential expressions, please refer to the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

FAQs

Q: What is the rule for simplifying exponential expressions? A: The rule for simplifying exponential expressions is to multiply the exponents when raising a power to another power.

Q: How do I simplify an expression with negative exponents? A: To simplify an expression with negative exponents, you can rewrite the negative exponent as a positive exponent by flipping the base.

Q: What is the difference between a power and an exponent? A: A power is the result of raising a base to a certain exponent. An exponent is the number that is raised to the power.

Q: How do I evaluate an expression with multiple exponents? A: To evaluate an expression with multiple exponents, you can apply the rules of exponents and simplify the expression step by step.