Write − 7 \sqrt{-7} − 7 ​ In Simplest Radical Form.

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Introduction

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When dealing with square roots of negative numbers, we often encounter the imaginary unit, denoted by $i$. In this case, we are tasked with simplifying the expression $\sqrt{-7}$ into its simplest radical form. To do this, we need to understand the properties of square roots and the concept of imaginary numbers.

Understanding Square Roots

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The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. However, when dealing with negative numbers, we need to introduce the imaginary unit, $i$, which is defined as the square root of -1.

The Imaginary Unit

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The imaginary unit, $i$, is a fundamental concept in mathematics that allows us to extend the real number system to the complex number system. It is defined as:

$$i = \sqrt{-1}$$

This means that $i^2 = -1$, which is a crucial property that we will use later to simplify the expression $\sqrt{-7}$.

Simplifying $\sqrt{-7}$

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To simplify $\sqrt{-7}$, we can use the property of the imaginary unit, $i$. We can rewrite $\sqrt{-7}$ as:

$$\sqrt{-7} = \sqrt{-1 \times 7}$$

Using the property of $i$, we can rewrite this as:

$$\sqrt{-7} = i\sqrt{7}$$

This is the simplest radical form of $\sqrt{-7}$, where we have factored out the imaginary unit, $i$, and left the square root of 7 inside.

Conclusion

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In this article, we have shown how to simplify the expression $\sqrt{-7}$ into its simplest radical form. We have used the properties of square roots and the concept of imaginary numbers to arrive at the final answer. The key takeaway is that when dealing with square roots of negative numbers, we need to introduce the imaginary unit, $i$, and use its properties to simplify the expression.

Final Answer

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The final answer is: $\boxed{i\sqrt{7}}$

Related Topics

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• Simplifying square roots of negative numbers
• Introduction to imaginary numbers
• Properties of the imaginary unit, $i$

Further Reading

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For more information on simplifying square roots of negative numbers and the concept of imaginary numbers, we recommend the following resources:

• Wikipedia: Imaginary unit
• Khan Academy: Imaginary numbers
• Mathway: Simplifying square roots of negative numbers

Note: The above article is a rewritten version of the original content, optimized for readability and SEO. The final answer and related topics are included to provide additional context and resources for readers.

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Introduction

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In our previous article, we explored how to simplify the expression $\sqrt{-7}$ into its simplest radical form. We introduced the concept of imaginary numbers and used the properties of the imaginary unit, $i$, to arrive at the final answer. In this article, we will address some common questions and concerns related to simplifying square roots of negative numbers.

Q&A

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Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed on the number line, such as 3, 4, or -2. An imaginary number, on the other hand, is a number that cannot be expressed on the number line, such as $i$, $\sqrt{-1}$, or $2i$.

Q: Why do we need to introduce the imaginary unit, $i$?

A: We need to introduce the imaginary unit, $i$, to extend the real number system to the complex number system. This allows us to work with square roots of negative numbers and other complex numbers.

Q: How do we simplify square roots of negative numbers?

A: To simplify square roots of negative numbers, we can use the property of the imaginary unit, $i$. We can rewrite $\sqrt{-7}$ as $i\sqrt{7}$, for example.

Q: Can we simplify square roots of negative numbers with a variable?

A: Yes, we can simplify square roots of negative numbers with a variable. For example, $\sqrt{-x}$ can be rewritten as $i\sqrt{x}$.

Q: What is the relationship between the square root of a negative number and the imaginary unit?

A: The square root of a negative number is equal to the imaginary unit multiplied by the square root of the absolute value of the number. For example, $\sqrt{-7} = i\sqrt{7}$.

Q: Can we simplify square roots of negative numbers with a fraction?

A: Yes, we can simplify square roots of negative numbers with a fraction. For example, $\sqrt{-\frac{1}{2}}$ can be rewritten as $i\sqrt{\frac{1}{2}}$.

Q: What is the final answer for $\sqrt{-7}$?

A: The final answer for $\sqrt{-7}$ is $i\sqrt{7}$.

Conclusion

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In this article, we have addressed some common questions and concerns related to simplifying square roots of negative numbers. We have introduced the concept of imaginary numbers and used the properties of the imaginary unit, $i$, to simplify expressions. We hope that this article has provided a clear understanding of this important topic.

Final Answer

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The final answer is: $i\sqrt{7}$

Related Topics

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• Simplifying square roots of negative numbers
• Introduction to imaginary numbers
• Properties of the imaginary unit, $i$

Further Reading

---

For more information on simplifying square roots of negative numbers and the concept of imaginary numbers, we recommend the following resources:

• Wikipedia: Imaginary unit
• Khan Academy: Imaginary numbers
• Mathway: Simplifying square roots of negative numbers

Note: The above article is a rewritten version of the original content, optimized for readability and SEO. The final answer and related topics are included to provide additional context and resources for readers.