Which Expression Is Equivalent To $24^{\frac{1}{3}}$?A. $2 \sqrt{3}$ B. \$2 \sqrt[3]{3}$[/tex\] C. $2 \sqrt{6}$ D. $2 \sqrt[3]{6}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $24^{\frac{1}{3}}$ and determining which of the given options is equivalent to it.

Understanding the Expression

The given expression is $24^{\frac{1}{3}}$. This expression represents the cube root of 24. To simplify this expression, we need to find the value that, when cubed, gives us 24.

Breaking Down the Expression

To simplify the expression, we can break it down into its prime factors. The prime factorization of 24 is $2^3 \times 3$. Now, we can rewrite the expression as $\left(2^3 \times 3\right)^{\frac{1}{3}}$.

Simplifying the Expression

Using the properties of exponents, we can simplify the expression further. When we have a power raised to another power, we can multiply the exponents. In this case, we have $\left(2^3 \times 3\right)^{\frac{1}{3}} = 2^{\frac{3}{3}} \times 3^{\frac{1}{3}}$.

Simplifying the Exponents

Now, we can simplify the exponents. $2^{\frac{3}{3}}$ is equal to $2^1$, which is simply 2. $3^{\frac{1}{3}}$ is the cube root of 3.

Combining the Simplified Exponents

Now, we can combine the simplified exponents to get the final result. $2 \times 3^{\frac{1}{3}}$ is equal to $2 \sqrt[3]{3}$.

Evaluating the Options

Now that we have simplified the expression, we can evaluate the given options to determine which one is equivalent to $24^{\frac{1}{3}}$.

Option A: $2 \sqrt{3}$

This option is not equivalent to $24^{\frac{1}{3}}$. The square root of 3 is not the same as the cube root of 3.

Option B: $2 \sqrt[3]{3}$

This option is equivalent to $24^{\frac{1}{3}}$. We simplified the expression to $2 \sqrt[3]{3}$, which is the same as this option.

Option C: $2 \sqrt{6}$

This option is not equivalent to $24^{\frac{1}{3}}$. The square root of 6 is not the same as the cube root of 3.

Option D: $2 \sqrt[3]{6}$

This option is not equivalent to $24^{\frac{1}{3}}$. The cube root of 6 is not the same as the cube root of 3.

Conclusion

In conclusion, the expression $24^{\frac{1}{3}}$ is equivalent to $2 \sqrt[3]{3}$. This is the correct answer among the given options. Understanding how to simplify radical expressions is crucial for solving various mathematical problems, and this article provides a step-by-step guide on how to simplify the expression $24^{\frac{1}{3}}$.

Frequently Asked Questions

Q: What is the cube root of 24?

A: The cube root of 24 is $24^{\frac{1}{3}}$.

Q: How do I simplify the expression $24^{\frac{1}{3}}$?

A: To simplify the expression, you can break it down into its prime factors and then use the properties of exponents to simplify it further.

Q: What is the equivalent expression to $24^{\frac{1}{3}}$?

A: The equivalent expression to $24^{\frac{1}{3}}$ is $2 \sqrt[3]{3}$.

References

Glossary

  • Radical expression: An expression that contains a radical sign, such as $\sqrt{x}$ or $\sqrt[3]{x}$.
  • Cube root: The cube root of a number is a value that, when cubed, gives us the original number.
  • Prime factorization: The process of breaking down a number into its prime factors.
    Frequently Asked Questions: Simplifying Radical Expressions ===========================================================

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will provide answers to frequently asked questions about simplifying radical expressions.

Q&A

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical sign, such as $\sqrt{x}$ or $\sqrt[3]{x}$.

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

A: A square root is the value that, when squared, gives us the original number, while a cube root is the value that, when cubed, gives us the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can break it down into its prime factors and then use the properties of exponents to simplify it further.

Q: What is the property of exponents that I can use to simplify radical expressions?

A: The property of exponents that you can use to simplify radical expressions is $\left(a^m \times bm\right)n = a^{m \times n} \times b^{m \times n}$.

Q: How do I simplify an expression like $\sqrt{24}$?

A: To simplify an expression like $\sqrt{24}$, you can break it down into its prime factors and then simplify it further using the properties of exponents.

Q: What is the simplified form of $\sqrt{24}$?

A: The simplified form of $\sqrt{24}$ is $2 \sqrt{6}$.

Q: How do I simplify an expression like $\sqrt[3]{24}$?

A: To simplify an expression like $\sqrt[3]{24}$, you can break it down into its prime factors and then simplify it further using the properties of exponents.

Q: What is the simplified form of $\sqrt[3]{24}$?

A: The simplified form of $\sqrt[3]{24}$ is $2 \sqrt[3]{3}$.

Q: What is the difference between a rational expression and a radical expression?

A: A rational expression is an expression that contains a fraction, while a radical expression is an expression that contains a radical sign.

Q: Can I simplify a radical expression that contains a rational expression?

A: Yes, you can simplify a radical expression that contains a rational expression by first simplifying the rational expression and then simplifying the radical expression.

Q: How do I simplify a radical expression that contains a rational expression?

A: To simplify a radical expression that contains a rational expression, you can first simplify the rational expression and then simplify the radical expression using the properties of exponents.

Conclusion

In conclusion, simplifying radical expressions is a crucial concept in mathematics, and understanding how to simplify them is essential for solving various mathematical problems. This article provides answers to frequently asked questions about simplifying radical expressions, and we hope that it will be helpful to you.

Frequently Asked Questions (FAQs)

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is an expression that contains a radical sign, while a rational expression is an expression that contains a fraction.

Q: Can I simplify a radical expression that contains a rational expression?

A: Yes, you can simplify a radical expression that contains a rational expression by first simplifying the rational expression and then simplifying the radical expression.

Q: How do I simplify a radical expression that contains a rational expression?

A: To simplify a radical expression that contains a rational expression, you can first simplify the rational expression and then simplify the radical expression using the properties of exponents.

References

Glossary

  • Radical expression: An expression that contains a radical sign, such as $\sqrt{x}$ or $\sqrt[3]{x}$.
  • Cube root: The cube root of a number is a value that, when cubed, gives us the original number.
  • Prime factorization: The process of breaking down a number into its prime factors.
  • Rational expression: An expression that contains a fraction.
  • Exponent: A small number that is raised to a power, such as $2^3$.
  • Property of exponents: A rule that is used to simplify expressions that contain exponents, such as $\left(a^m \times bm\right)n = a^{m \times n} \times b^{m \times n}$.

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