Which Of The Following Is A Like Radical To $\sqrt[3]{7x}$?A. $4\sqrt[3]{7x}$B. $\sqrt{7x}$C. $x\sqrt[3]{7}$D. $7\sqrt{x}$

Introduction to Like Radicals

In mathematics, like radicals are expressions that contain the same radical part. When dealing with radicals, it's essential to understand the concept of like radicals to simplify expressions and perform operations. In this article, we will explore the concept of like radicals and determine which of the given options is a like radical to $\sqrt[3]{7x}$.

What are Like Radicals?

Like radicals are expressions that contain the same radical part. For example, $\sqrt{2}$ and $\sqrt{8}$ are like radicals because they both contain the radical part $\sqrt{2}$. Similarly, $\sqrt[3]{4}$ and $\sqrt[3]{64}$ are like radicals because they both contain the radical part $\sqrt[3]{4}$.

Properties of Like Radicals

Like radicals have several properties that make them useful in mathematics. Some of the key properties of like radicals include:

• Multiplication: When multiplying like radicals, we can multiply the coefficients (numbers in front of the radical) and keep the radical part the same. For example, $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$.
• Division: When dividing like radicals, we can divide the coefficients and keep the radical part the same. For example, $\sqrt{8} \div \sqrt{2} = \sqrt{4} = 2$.
• Addition and Subtraction: When adding or subtracting like radicals, we can add or subtract the coefficients and keep the radical part the same. For example, $\sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$.

Determining Like Radicals

To determine if two expressions are like radicals, we need to compare their radical parts. If the radical parts are the same, then the expressions are like radicals.

Example: Finding a Like Radical to $\sqrt[3]{7x}$

Now, let's find a like radical to $\sqrt[3]{7x}$. To do this, we need to compare the radical part of $\sqrt[3]{7x}$ with the radical parts of the given options.

Option A: $4\sqrt[3]{7x}$

The radical part of $4\sqrt[3]{7x}$ is $\sqrt[3]{7x}$, which is the same as the radical part of $\sqrt[3]{7x}$. Therefore, $4\sqrt[3]{7x}$ is a like radical to $\sqrt[3]{7x}$.

Option B: $\sqrt{7x}$

The radical part of $\sqrt{7x}$ is $\sqrt{7x}$, which is not the same as the radical part of $\sqrt[3]{7x}$. Therefore, $\sqrt{7x}$ is not a like radical to $\sqrt[3]{7x}$.

Option C: $x\sqrt[3]{7}$

The radical part of $x\sqrt[3]{7}$ is $\sqrt[3]{7}$, which is not the same as the radical part of $\sqrt[3]{7x}$. Therefore, $x\sqrt[3]{7}$ is not a like radical to $\sqrt[3]{7x}$.

Option D: $7\sqrt{x}$

The radical part of $7\sqrt{x}$ is $\sqrt{x}$, which is not the same as the radical part of $\sqrt[3]{7x}$. Therefore, $7\sqrt{x}$ is not a like radical to $\sqrt[3]{7x}$.

Conclusion

In conclusion, the only option that is a like radical to $\sqrt[3]{7x}$ is $4\sqrt[3]{7x}$. This is because the radical part of $4\sqrt[3]{7x}$ is $\sqrt[3]{7x}$, which is the same as the radical part of $\sqrt[3]{7x}$.

Final Answer

The final answer is: $\boxed{4\sqrt[3]{7x}}$

Introduction to Like Radicals Q&A

In our previous article, we explored the concept of like radicals in mathematics. Like radicals are expressions that contain the same radical part. In this article, we will answer some frequently asked questions about like radicals to help you better understand the concept.

Q: What is the difference between like radicals and unlike radicals?

A: Like radicals are expressions that contain the same radical part, while unlike radicals are expressions that contain different radical parts. For example, $\sqrt{2}$ and $\sqrt{8}$ are like radicals because they both contain the radical part $\sqrt{2}$, while $\sqrt{2}$ and $\sqrt{3}$ are unlike radicals because they contain different radical parts.

Q: How do I determine if two expressions are like radicals?

A: To determine if two expressions are like radicals, you need to compare their radical parts. If the radical parts are the same, then the expressions are like radicals.

Q: Can I add or subtract like radicals?

A: Yes, you can add or subtract like radicals. When adding or subtracting like radicals, you can add or subtract the coefficients (numbers in front of the radical) and keep the radical part the same. For example, $\sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$.

Q: Can I multiply or divide like radicals?

A: Yes, you can multiply or divide like radicals. When multiplying like radicals, you can multiply the coefficients and keep the radical part the same. For example, $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$. When dividing like radicals, you can divide the coefficients and keep the radical part the same. For example, $\sqrt{8} \div \sqrt{2} = \sqrt{4} = 2$.

Q: What is the relationship between like radicals and the distributive property?

A: The distributive property is a mathematical concept that allows you to distribute a coefficient to multiple terms. Like radicals are related to the distributive property in that you can use the distributive property to simplify expressions that contain like radicals. For example, $3(\sqrt{2} + \sqrt{8}) = 3\sqrt{2} + 3\sqrt{8}$.

Q: Can I simplify expressions that contain like radicals?

A: Yes, you can simplify expressions that contain like radicals. When simplifying expressions that contain like radicals, you can combine the coefficients and keep the radical part the same. For example, $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.

Q: What are some common mistakes to avoid when working with like radicals?

A: Some common mistakes to avoid when working with like radicals include:

• Not comparing the radical parts: Make sure to compare the radical parts of the expressions to determine if they are like radicals.
• Not using the distributive property: Use the distributive property to simplify expressions that contain like radicals.
• Not combining like radicals: Combine like radicals to simplify expressions.

Conclusion

In conclusion, like radicals are an essential concept in mathematics that allows you to simplify expressions and perform operations. By understanding the concept of like radicals and how to apply it, you can become a more confident and proficient mathematician.

Final Tips

• Practice, practice, practice: The more you practice working with like radicals, the more comfortable you will become with the concept.
• Use real-world examples: Use real-world examples to help you understand the concept of like radicals and how to apply it.
• Seek help when needed: Don't be afraid to seek help when you need it. Ask your teacher or tutor for assistance, or seek help online.

Final Answer

The final answer is: Like radicals are expressions that contain the same radical part, and they can be added, subtracted, multiplied, and divided just like regular numbers.