Convert The Expression To Radical Form.$\[ X^{\frac{4}{3}} Y^{\frac{1}{3}} Z^{\frac{2}{3}} \\]A. \[$\sqrt[3]{x^4 Y Z^2}\$\]B. \[$\sqrt[4]{x^3 Y Z^3}\$\]C. \[$\sqrt[3]{x^4 Y Z}\$\]D. \[$\sqrt{x^4 Y Z^2}\$\]

Introduction

Radical expressions are a fundamental concept in mathematics, and converting expressions to radical form is a crucial skill for students to master. In this article, we will explore the process of converting expressions to radical form, with a focus on the given expression: $x^{\frac{4}{3}} y^{\frac{1}{3}} z^{\frac{2}{3}}$. We will examine the different options provided and determine the correct radical form of the expression.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. The most common radical expression is the square root, denoted by $\sqrt{x}$. However, there are other types of radical expressions, such as cube roots, fourth roots, and so on. The general form of a radical expression is:

$$\sqrt[n]{x}$$

where $n$ is the index of the radical and $x$ is the radicand.

Converting Expressions to Radical Form

To convert an expression to radical form, we need to identify the index of the radical and the radicand. In the given expression, $x^{\frac{4}{3}} y^{\frac{1}{3}} z^{\frac{2}{3}}$, we can see that the index of the radical is $\frac{4}{3}$, $\frac{1}{3}$, and $\frac{2}{3}$ for $x$, $y$, and $z$ respectively. The radicand is the expression inside the radical sign.

To convert the expression to radical form, we can use the following steps:

1. Identify the index of the radical for each variable.
2. Rewrite each variable as a power of the index.
3. Combine the powers of the variables using the product rule of exponents.

Applying the Steps to the Given Expression

Let's apply the steps to the given expression:

1. Identify the index of the radical for each variable:

$$x^{\frac{4}{3}} \implies \sqrt[3]{x^4}$$

$$y^{\frac{1}{3}} \implies \sqrt[3]{y}$$

$$z^{\frac{2}{3}} \implies \sqrt[3]{z^2}$$

2. Rewrite each variable as a power of the index:

$$\sqrt[3]{x^4} = x^{\frac{4}{3}}$$

$$\sqrt[3]{y} = y^{\frac{1}{3}}$$

$$\sqrt[3]{z^2} = z^{\frac{2}{3}}$$

3. Combine the powers of the variables using the product rule of exponents:

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

Evaluating the Options

Now that we have converted the expression to radical form, let's evaluate the options provided:

A. $\sqrt[3]{x^4 y z^2}$

B. $\sqrt[4]{x^3 y z^3}$

C. $\sqrt[3]{x^4 y z}$

D. $\sqrt{x^4 y z^2}$

Based on our analysis, option A is the correct radical form of the expression.

Conclusion

Converting expressions to radical form is an essential skill for students to master. By following the steps outlined in this article, we can convert expressions to radical form and evaluate the options provided. In this case, the correct radical form of the expression $x^{\frac{4}{3}} y^{\frac{1}{3}} z^{\frac{2}{3}}$ is $\sqrt[3]{x^4 y z^2}$. We hope that this article has provided a comprehensive guide to converting expressions to radical form and has helped students to better understand this important mathematical concept.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Radical Expressions and Equations
• Mathway: Radical Expressions and Equations
• IXL: Radical Expressions and Equations

Introduction

In our previous article, we explored the process of converting expressions to radical form, with a focus on the given expression: $x^{\frac{4}{3}} y^{\frac{1}{3}} z^{\frac{2}{3}}$. We also evaluated the options provided and determined the correct radical form of the expression. In this article, we will provide a Q&A guide to help students better understand the concept of converting expressions to radical form.

Q&A Guide

Q: What is the index of the radical?

A: The index of the radical is the number that is written outside the radical sign. For example, in the expression $\sqrt[3]{x}$, the index of the radical is 3.

Q: What is the radicand?

A: The radicand is the expression inside the radical sign. For example, in the expression $\sqrt[3]{x}$, the radicand is $x$.

Q: How do I convert an expression to radical form?

A: To convert an expression to radical form, you need to identify the index of the radical and the radicand. Then, you can use the following steps:

1. Rewrite each variable as a power of the index.
2. Combine the powers of the variables using the product rule of exponents.

Q: What is the product rule of exponents?

A: The product rule of exponents states that when you multiply two or more variables with the same base, you can add their exponents. For example, $x^a \cdot x^b = x^{a+b}$.

Q: How do I evaluate the options provided?

A: To evaluate the options provided, you need to compare the options with the correct radical form of the expression. You can use the following steps:

1. Identify the index of the radical and the radicand in each option.
2. Compare the options with the correct radical form of the expression.
3. Select the option that matches the correct radical form of the expression.

Q: What are some common mistakes to avoid when converting expressions to radical form?

A: Some common mistakes to avoid when converting expressions to radical form include:

• Forgetting to identify the index of the radical and the radicand.
• Not using the product rule of exponents to combine the powers of the variables.
• Not evaluating the options provided carefully.

Q: How can I practice converting expressions to radical form?

A: You can practice converting expressions to radical form by using the following resources:

• Khan Academy: Radical Expressions and Equations
• Mathway: Radical Expressions and Equations
• IXL: Radical Expressions and Equations

Conclusion

Converting expressions to radical form is an essential skill for students to master. By following the steps outlined in this article and practicing with the resources provided, students can improve their skills in converting expressions to radical form and better understand this important mathematical concept.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Radical Expressions and Equations
• Mathway: Radical Expressions and Equations
• IXL: Radical Expressions and Equations

By practicing and reviewing these resources, students can improve their skills in converting expressions to radical form and better understand this important mathematical concept.