What Is The Simplified Form Of The Following Expression? Assume $x \ \textgreater \ 0$.A. $\sqrt[4]{\frac{3}{2 X}}$B. $\frac{\sqrt[4]{6 X}}{2 X}$C. $\frac{\sqrt[4]{24 X^3}}{2 X}$D. $\frac{\sqrt[4]{24 X^3}}{16

Introduction

When dealing with mathematical expressions, simplifying them is an essential step to understand their behavior and make calculations easier. In this article, we will focus on simplifying a given expression that involves radicals and fractions. We will assume that the variable $x$ is greater than $0$, which will help us in simplifying the expression.

Understanding the Expression

The given expression is $\sqrt[4]{\frac{3}{2 x}}$. To simplify this expression, we need to understand the properties of radicals and fractions. The expression involves a fourth root, which means that we need to find the value that, when raised to the fourth power, gives us the expression inside the radical.

Simplifying the Expression

To simplify the expression, we can start by rationalizing the denominator. This involves multiplying the numerator and denominator by a value that will eliminate the radical in the denominator. In this case, we can multiply the numerator and denominator by $\sqrt[4]{2 x}$.

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

Simplifying the Radical

Now, we can simplify the radical by combining the two terms in the numerator. This will give us a single term with a fourth root.

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

Canceling Out the Common Terms

Now, we can cancel out the common terms in the numerator and denominator. This will give us a simplified expression.

\frac{\sqrt[4]{3 \cdot 2 x}}{\sqrt[4]{2 x} \cdot \sqrt[4]{2 x}} = \frac{\sqrt[4]{6 x}}{2 x}

Conclusion

In conclusion, the simplified form of the given expression is $\frac{\sqrt[4]{6 x}}{2 x}$. This expression is obtained by rationalizing the denominator and simplifying the radical.

Comparison with the Options

Now, let's compare the simplified expression with the options given in the problem.

• Option A: $\sqrt[4]{\frac{3}{2 x}}$ is the original expression, not the simplified form.
• Option B: $\frac{\sqrt[4]{6 x}}{2 x}$ is the simplified form of the expression.
• Option C: $\frac{\sqrt[4]{24 x^3}}{2 x}$ is not the simplified form of the expression.
• Option D: $\frac{\sqrt[4]{24 x^3}}{16}$ is not the simplified form of the expression.

Final Answer

The final answer is $\boxed{\frac{\sqrt[4]{6 x}}{2 x}}$.

Introduction

In our previous article, we discussed the simplified form of a given mathematical expression. In this article, we will address some frequently asked questions (FAQs) related to simplifying mathematical expressions. We will cover topics such as rationalizing the denominator, simplifying radicals, and canceling out common terms.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying the numerator and denominator by a value that will eliminate the radical in the denominator. This is done to simplify the expression and make it easier to work with.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply the numerator and denominator by a value that will eliminate the radical in the denominator. For example, if you have an expression like $\frac{\sqrt{2}}{2}$, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$.

Q: What is simplifying a radical?

A: Simplifying a radical involves finding the value that, when raised to the power of the index of the radical, gives us the expression inside the radical. For example, if we have an expression like $\sqrt{12}$, we can simplify it by finding the value that, when squared, gives us $12$.

Q: How do I simplify a radical?

A: To simplify a radical, you need to find the value that, when raised to the power of the index of the radical, gives us the expression inside the radical. For example, if we have an expression like $\sqrt{12}$, we can simplify it by finding the value that, when squared, gives us $12$. In this case, the value is $2\sqrt{3}$.

Q: What is canceling out common terms?

A: Canceling out common terms involves eliminating common factors in the numerator and denominator of an expression. This is done to simplify the expression and make it easier to work with.

Q: How do I cancel out common terms?

A: To cancel out common terms, you need to identify the common factors in the numerator and denominator of an expression and eliminate them. For example, if you have an expression like $\frac{2x}{2x}$, you can cancel out the common term $2x$.

Q: What are some common mistakes to avoid when simplifying mathematical expressions?

A: Some common mistakes to avoid when simplifying mathematical expressions include:

• Not rationalizing the denominator
• Not simplifying radicals
• Not canceling out common terms
• Making errors when multiplying or dividing expressions

Q: How can I practice simplifying mathematical expressions?

A: You can practice simplifying mathematical expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and checking your work with a calculator or online tool.

Q: What are some real-world applications of simplifying mathematical expressions?

A: Simplifying mathematical expressions has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and perimeter of shapes
• Solving equations and inequalities
• Modeling real-world phenomena

Conclusion

In conclusion, simplifying mathematical expressions is an essential skill that has many real-world applications. By understanding how to rationalize the denominator, simplify radicals, and cancel out common terms, you can make calculations easier and more efficient. We hope this article has been helpful in addressing some frequently asked questions related to simplifying mathematical expressions.