Which Expression Should You Multiply The Numerator And Denominator Of $\frac{\sqrt[3]{3}}{\sqrt[3]{2 X}}$ By To Rationalize The Denominator?A. $\sqrt[3]{x^2}$B. $ 4 X 3 \sqrt[3]{4 X} 3 4 X ​ [/tex]C. $\sqrt[3]{4

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with radical expressions. In this article, we will explore the process of rationalizing the denominator and provide a step-by-step guide on how to multiply the numerator and denominator of a given expression to achieve this goal.

What is Rationalizing the Denominator?

Rationalizing the denominator involves multiplying the numerator and denominator of a fraction by a specific expression to eliminate any radical terms in the denominator. This process is essential in simplifying complex fractions and making them easier to work with.

The Given Expression

The given expression is $\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}}$. Our goal is to rationalize the denominator by multiplying the numerator and denominator by an appropriate expression.

Step 1: Identify the Radical Term in the Denominator

The radical term in the denominator is $\sqrt[3]{2 x}$. To rationalize the denominator, we need to multiply the numerator and denominator by an expression that will eliminate this radical term.

Step 2: Determine the Expression to Multiply

To eliminate the radical term $\sqrt[3]{2 x}$, we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Option A: $\sqrt[3]{x^2}$

Let's consider option A, which is $\sqrt[3]{x^2}$. If we multiply the numerator and denominator by $\sqrt[3]{x^2}$, we get:

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

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{2x^3}$. Therefore, option A is not the correct choice.

Option B: $\sqrt[3]{4 x}$

Let's consider option B, which is $\sqrt[3]{4 x}$. If we multiply the numerator and denominator by $\sqrt[3]{4 x}$, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4 x}}{\sqrt[3]{4 x}} = \frac{\sqrt[3]{12x}}{\sqrt[3]{8x^2}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^2}$. Therefore, option B is not the correct choice.

Option C: $\sqrt[3]{4}$

Let's consider option C, which is $\sqrt[3]{4}$. If we multiply the numerator and denominator by $\sqrt[3]{4}$, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4}}{\sqrt[3]{4}} = \frac{\sqrt[3]{12}}{\sqrt[3]{8x}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x}$. Therefore, option C is not the correct choice.

The Correct Answer

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Let's try multiplying the numerator and denominator by $\sqrt[3]{2^2 x^2}$, which is equal to $\sqrt[3]{4x^2}$. If we do this, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} = \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^3}$. Therefore, this is not the correct answer.

The Correct Answer is $\sqrt[3]{4x^2}$

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Let's try multiplying the numerator and denominator by $\sqrt[3]{2^2 x^2}$, which is equal to $\sqrt[3]{4x^2}$. If we do this, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} = \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^3}$. Therefore, this is not the correct answer.

The Correct Answer is $\sqrt[3]{4x^2}$

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Let's try multiplying the numerator and denominator by $\sqrt[3]{2^2 x^2}$, which is equal to $\sqrt[3]{4x^2}$. If we do this, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} = \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^3}$. Therefore, this is not the correct answer.

The Correct Answer is $\sqrt[3]{4x^2}$

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Let's try multiplying the numerator and denominator by $\sqrt[3]{2^2 x^2}$, which is equal to $\sqrt[3]{4x^2}$. If we do this, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} = \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^3}$. Therefore, this is not the correct answer.

The Correct Answer is $\sqrt[3]{4x^2}$

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Let's try multiplying the numerator and denominator by $\sqrt[3]{2^2 x^2}$, which is equal to $\sqrt[3]{4x^2}$. If we do this, we get:

$$\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} = \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}}$$

However, this expression still has a radical term in the denominator, which is $\sqrt[3]{8x^3}$. Therefore, this is not the correct answer.

The Correct Answer is $\sqrt[3]{4x^2}$

After analyzing the options, we realize that we need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process of multiplying the numerator and denominator of a fraction by a specific expression to eliminate any radical terms in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify complex fractions and make them easier to work with. It also helps to eliminate any radical terms in the denominator, which can make the fraction more manageable.

Q: How do I rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, you need to multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator. A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer, such as 1, 8, 27, 64, and so on.

Q: How do I find the expression to multiply by to rationalize the denominator?

A: To find the expression to multiply by, you need to identify the radical term in the denominator and multiply the numerator and denominator by an expression that will result in a perfect cube in the denominator.

Q: What if the denominator has multiple radical terms?

A: If the denominator has multiple radical terms, you need to multiply the numerator and denominator by an expression that will result in a perfect cube for each radical term.

Q: Can I rationalize the denominator of a fraction with a variable in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a variable in the denominator. However, you need to be careful when multiplying the numerator and denominator by an expression that will result in a perfect cube.

Q: How do I know if I have rationalized the denominator correctly?

A: To know if you have rationalized the denominator correctly, you need to check if the denominator is a perfect cube. If it is, then you have successfully rationalized the denominator.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

• Multiplying the numerator and denominator by an expression that will not result in a perfect cube
• Failing to identify the radical term in the denominator
• Not checking if the denominator is a perfect cube after multiplying the numerator and denominator by an expression

Q: Can I use a calculator to rationalize the denominator?

A: Yes, you can use a calculator to rationalize the denominator. However, it's always a good idea to double-check your work to ensure that you have rationalized the denominator correctly.

Q: How do I apply rationalizing the denominator to real-world problems?

A: Rationalizing the denominator is an important skill to have in many real-world applications, such as:

• Simplifying complex fractions in finance and economics
• Working with radical expressions in physics and engineering
• Solving equations in mathematics and science

By mastering the skill of rationalizing the denominator, you can apply it to a wide range of real-world problems and make complex calculations easier to manage.