Write $\sqrt[3]{\sqrt{n^3}}$ As A Single Radical Using The Smallest Possible Root.

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Introduction

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Radicals are a fundamental concept in mathematics, and understanding how to simplify and manipulate them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\sqrt[3]{\sqrt{n^3}}$ as a single radical using the smallest possible root. We will break down the problem step by step, using mathematical concepts and formulas to arrive at the final solution.

Understanding Radicals

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A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and a prime number. The radical symbol, denoted by $\sqrt[n]{x}$, represents the nth root of a number x. For example, $\sqrt[3]{8}$ represents the cube root of 8, which is equal to 2.

Simplifying the Expression

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To simplify the expression $\sqrt[3]{\sqrt{n^3}}$, we need to start by understanding the properties of radicals. One of the key properties of radicals is that the nth root of a number can be expressed as the product of the (n-1)th root of the number and the nth root of the number.

Using this property, we can rewrite the expression $\sqrt[3]{\sqrt{n^3}}$ as follows:

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

Applying the Property of Radicals

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Now, we can apply the property of radicals to simplify the expression further. We know that the nth root of a number can be expressed as the product of the (n-1)th root of the number and the nth root of the number. Therefore, we can rewrite the expression as follows:

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

Simplifying the Fraction

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Now, we can simplify the fraction inside the radical. We know that the square root of a number can be expressed as the product of the number and the square root of the number. Therefore, we can rewrite the fraction as follows:

$$\sqrt[3]{\frac{\sqrt{n^3}}{\sqrt{n^3}}} = \sqrt[3]{\frac{\sqrt{n^3}}{\sqrt{n^3}}} = \sqrt[3]{\frac{n^3}{n^3}} = \sqrt[3]{1}$$

Evaluating the Radical

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Now, we can evaluate the radical. We know that the cube root of 1 is equal to 1. Therefore, we can rewrite the expression as follows:

$$\sqrt[3]{\sqrt{n^3}} = \sqrt[3]{\sqrt{n^3}} \cdot \sqrt[3]{\frac{\sqrt{n^3}}{\sqrt{n^3}}} = \sqrt[3]{\sqrt{n^3}} \cdot \sqrt[3]{1} = \sqrt[3]{\sqrt{n^3}} \cdot 1 = \sqrt[3]{\sqrt{n^3}}$$

Conclusion

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In conclusion, we have simplified the expression $\sqrt[3]{\sqrt{n^3}}$ as a single radical using the smallest possible root. We have used mathematical concepts and formulas to arrive at the final solution. The expression can be simplified as follows:

$$\sqrt[3]{\sqrt{n^3}} = \sqrt[3]{\sqrt{n^3}} = \sqrt[3]{n}$$

Final Answer

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The final answer is $\boxed{\sqrt[3]{n}}$.

Frequently Asked Questions

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Q: What is the smallest possible root for the expression $\sqrt[3]{\sqrt{n^3}}$?

A: The smallest possible root for the expression $\sqrt[3]{\sqrt{n^3}}$ is 3.

Q: How do you simplify the expression $\sqrt[3]{\sqrt{n^3}}$?

A: To simplify the expression $\sqrt[3]{\sqrt{n^3}}$, you can use the property of radicals that the nth root of a number can be expressed as the product of the (n-1)th root of the number and the nth root of the number.

Q: What is the final answer for the expression $\sqrt[3]{\sqrt{n^3}}$?

A: The final answer for the expression $\sqrt[3]{\sqrt{n^3}}$ is $\boxed{\sqrt[3]{n}}$.

References

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• [1] "Radicals" by Math Open Reference. Retrieved February 26, 2024.
• [2] "Properties of Radicals" by Purplemath. Retrieved February 26, 2024.
• [3] "Simplifying Radicals" by Mathway. Retrieved February 26, 2024.
  

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Introduction

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In our previous article, we discussed how to simplify the expression $\sqrt[3]{\sqrt{n^3}}$ as a single radical using the smallest possible root. In this article, we will answer some frequently asked questions about simplifying radicals.

Q&A

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Q: What is the smallest possible root for the expression $\sqrt[3]{\sqrt{n^3}}$?

A: The smallest possible root for the expression $\sqrt[3]{\sqrt{n^3}}$ is 3.

Q: How do you simplify the expression $\sqrt[3]{\sqrt{n^3}}$?

A: To simplify the expression $\sqrt[3]{\sqrt{n^3}}$, you can use the property of radicals that the nth root of a number can be expressed as the product of the (n-1)th root of the number and the nth root of the number.

Q: What is the final answer for the expression $\sqrt[3]{\sqrt{n^3}}$?

A: The final answer for the expression $\sqrt[3]{\sqrt{n^3}}$ is $\boxed{\sqrt[3]{n}}$.

Q: Can you simplify the expression $\sqrt[4]{\sqrt{16}}$?

A: Yes, you can simplify the expression $\sqrt[4]{\sqrt{16}}$ by using the property of radicals. The expression can be simplified as follows:

$$\sqrt[4]{\sqrt{16}} = \sqrt[4]{\sqrt{16}} = \sqrt[4]{4} = \sqrt[2]{2} = 2$$

Q: How do you simplify the expression $\sqrt[5]{\sqrt{32}}$?

A: To simplify the expression $\sqrt[5]{\sqrt{32}}$, you can use the property of radicals that the nth root of a number can be expressed as the product of the (n-1)th root of the number and the nth root of the number. The expression can be simplified as follows:

$$\sqrt[5]{\sqrt{32}} = \sqrt[5]{\sqrt{32}} = \sqrt[5]{\sqrt{16 \cdot 2}} = \sqrt[5]{\sqrt{16} \cdot \sqrt{2}} = \sqrt[5]{4 \cdot \sqrt{2}} = \sqrt[5]{4} \cdot \sqrt[5]{\sqrt{2}} = 2 \cdot \sqrt[5]{\sqrt{2}}$$

Q: Can you simplify the expression $\sqrt[6]{\sqrt{64}}$?

A: Yes, you can simplify the expression $\sqrt[6]{\sqrt{64}}$ by using the property of radicals. The expression can be simplified as follows:

$$\sqrt[6]{\sqrt{64}} = \sqrt[6]{\sqrt{64}} = \sqrt[6]{\sqrt{4^3}} = \sqrt[6]{4^3} = \sqrt[6]{4^3} = \sqrt[3]{4} = 2$$

Conclusion

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In conclusion, we have answered some frequently asked questions about simplifying radicals. We have used mathematical concepts and formulas to arrive at the final solutions. The expressions can be simplified as follows:

• $\sqrt[3]{\sqrt{n^3}} = \sqrt[3]{n}$
• $\sqrt[4]{\sqrt{16}} = 2$
• $\sqrt[5]{\sqrt{32}} = 2 \cdot \sqrt[5]{\sqrt{2}}$
• $\sqrt[6]{\sqrt{64}} = 2$

Final Answer

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The final answer is $\boxed{\sqrt[3]{n}}$.

References

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• [1] "Radicals" by Math Open Reference. Retrieved February 26, 2024.
• [2] "Properties of Radicals" by Purplemath. Retrieved February 26, 2024.
• [3] "Simplifying Radicals" by Mathway. Retrieved February 26, 2024.