Which Of The Following Is Equivalent To $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$?A. $\frac{2 \sqrt{a}}{a}$B. \$\frac{\sqrt[3]{4a^2}}{a}$[/tex\]C. $\frac{2 \sqrt[3]{a}}{a}$D. $\sqrt[3]{4}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$. We will examine each option and determine which one is equivalent to the given expression.

Understanding Radical Expressions

Before we dive into the simplification process, let's take a moment to understand what radical expressions are. A radical expression is any expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$. The radical sign indicates that the expression inside the sign is to be taken to the power of $1/n$.

Simplifying the Given Expression

Now that we have a basic understanding of radical expressions, let's focus on simplifying the given expression $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$. To simplify this expression, we can use the rule of radicals, which states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$.

Using this rule, we can rewrite the given expression as $\sqrt[3]{\frac{4a}{a^2}}$. Now, let's simplify the fraction inside the radical sign.

Simplifying the Fraction

To simplify the fraction $\frac{4a}{a^2}$, we can divide both the numerator and the denominator by $a$. This gives us $\frac{4}{a}$.

Now, let's substitute this simplified fraction back into the original expression. We have $\sqrt[3]{\frac{4}{a}}$.

Evaluating the Options

Now that we have simplified the given expression, let's evaluate each option to determine which one is equivalent.

Option A: $\frac{2 \sqrt{a}}{a}$

To evaluate this option, we need to simplify the expression inside the radical sign. We can rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$. Now, let's substitute this expression back into the original option.

We have $\frac{2 a^{\frac{1}{2}}}{a}$. Now, let's simplify the fraction by dividing both the numerator and the denominator by $a^{\frac{1}{2}}$. This gives us $\frac{2}{a^{\frac{1}{2}}}$.

Now, let's rewrite this expression using the rule of exponents, which states that $a^{\frac{1}{2}} = \sqrt{a}$. We have $\frac{2}{\sqrt{a}}$.

Option B: $\frac{\sqrt[3]{4a^2}}{a}$

To evaluate this option, we need to simplify the expression inside the radical sign. We can rewrite $\sqrt[3]{4a^2}$ as $\sqrt[3]{4} \cdot \sqrt[3]{a^2}$.

Now, let's substitute this expression back into the original option. We have $\frac{\sqrt[3]{4} \cdot \sqrt[3]{a^2}}{a}$.

Now, let's simplify the fraction by dividing both the numerator and the denominator by $a$. This gives us $\frac{\sqrt[3]{4} \cdot \sqrt[3]{a^2}}{a}$.

Option C: $\frac{2 \sqrt[3]{a}}{a}$

To evaluate this option, we need to simplify the expression inside the radical sign. We can rewrite $\sqrt[3]{a}$ as $a^{\frac{1}{3}}$.

Now, let's substitute this expression back into the original option. We have $\frac{2 a^{\frac{1}{3}}}{a}$.

Now, let's simplify the fraction by dividing both the numerator and the denominator by $a^{\frac{1}{3}}$. This gives us $\frac{2}{a^{\frac{2}{3}}}$.

Now, let's rewrite this expression using the rule of exponents, which states that $a^{\frac{2}{3}} = \sqrt[3]{a^2}$. We have $\frac{2}{\sqrt[3]{a^2}}$.

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

To evaluate this option, we need to simplify the expression inside the radical sign. We can rewrite $\sqrt[3]{4}$ as $\sqrt[3]{4}$.

Now, let's substitute this expression back into the original option. We have $\sqrt[3]{4}$.

Conclusion

In conclusion, we have evaluated each option and determined which one is equivalent to the given expression $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$. We found that option C, $\frac{2 \sqrt[3]{a}}{a}$, is the correct answer.

Final Answer

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$. We evaluated each option and determined which one is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand the process of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is any expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$. The radical sign indicates that the expression inside the sign is to be taken to the power of $1/n$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the rule of radicals, which states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$. You can also use the rule of exponents, which states that $a^{\frac{1}{n}} = \sqrt[n]{a}$.

Q: What is the difference between a cube root and a square root?

A: A cube root is a radical expression that has a power of $3$, while a square root is a radical expression that has a power of $2$. For example, $\sqrt[3]{x}$ is a cube root, while $\sqrt{x}$ is a square root.

Q: How do I simplify a fraction inside a radical sign?

A: To simplify a fraction inside a radical sign, you can divide both the numerator and the denominator by the greatest common factor (GCF). For example, $\sqrt{\frac{12}{16}}$ can be simplified by dividing both the numerator and the denominator by $4$, resulting in $\sqrt{\frac{3}{4}}$.

Q: What is the rule of exponents?

A: The rule of exponents states that $a^{\frac{1}{n}} = \sqrt[n]{a}$. This means that you can rewrite a radical expression as an exponent expression, and vice versa.

Q: How do I simplify an expression with multiple radical signs?

A: To simplify an expression with multiple radical signs, you can use the rule of radicals, which states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$. You can also use the rule of exponents, which states that $a^{\frac{1}{n}} = \sqrt[n]{a}$.

Q: What is the final answer to the given expression $\frac{\sqrt[3]{4a}}{\sqrt[3]{a^2}}$?

A: The final answer is option C, $\frac{2 \sqrt[3]{a}}{a}$.

Conclusion

In conclusion, we have provided a Q&A guide to help you better understand the process of simplifying radical expressions. We have covered topics such as the definition of a radical expression, the rule of radicals, and the rule of exponents. We have also provided examples and explanations to help you understand the concepts.

Final Tips

  • Always simplify radical expressions by using the rule of radicals and the rule of exponents.
  • Use the greatest common factor (GCF) to simplify fractions inside radical signs.
  • Rewrite radical expressions as exponent expressions, and vice versa.
  • Practice, practice, practice! Simplifying radical expressions takes practice, so be sure to try out different examples and exercises.

Additional Resources

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

Final Answer

The final answer is: C\boxed{C}