Write − 175 \sqrt{-175} − 175 ​ 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 $i$, which is defined as the square root of $-1$. In this article, we will explore how to write $\sqrt{-175}$ in its simplest radical form, which involves expressing it as a product of a real number and the imaginary unit $i$.

Understanding the Concept of Imaginary Numbers

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Before we dive into the problem, let's briefly review the concept of imaginary numbers. Imaginary numbers are a way to extend the real number system to include numbers that, when squared, give a negative result. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. This means that $i^2 = -1$.

Breaking Down the Problem

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To write $\sqrt{-175}$ in its simplest radical form, we need to break down the number $-175$ into its prime factors. This will help us identify any perfect squares that can be extracted from the expression.

Prime Factorization of -175

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The prime factorization of $-175$ is:

$$-175 = -1 \cdot 5^2 \cdot 7$$

Expressing $\sqrt{-175}$ in Terms of $i$

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Now that we have the prime factorization of $-175$, we can express $\sqrt{-175}$ in terms of $i$. We can rewrite $\sqrt{-175}$ as:

$$\sqrt{-175} = \sqrt{-1 \cdot 5^2 \cdot 7}$$

Simplifying the Expression

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Using the property of square roots that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we can simplify the expression as:

$$\sqrt{-175} = \sqrt{-1} \cdot \sqrt{5^2} \cdot \sqrt{7}$$

Introducing the Imaginary Unit $i$

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Since $\sqrt{-1} = i$, we can substitute this into the expression:

$$\sqrt{-175} = i \cdot 5 \cdot \sqrt{7}$$

Final Simplification

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The expression $i \cdot 5 \cdot \sqrt{7}$ is already in its simplest radical form. We cannot simplify it further, as there are no more perfect squares that can be extracted from the expression.

Conclusion

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In this article, we have shown how to write $\sqrt{-175}$ in its simplest radical form. We started by breaking down the number $-175$ into its prime factors, and then expressed $\sqrt{-175}$ in terms of the imaginary unit $i$. Finally, we simplified the expression to obtain the final answer.

Example Use Cases

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The concept of imaginary numbers and the ability to write $\sqrt{-175}$ in its simplest radical form have numerous applications in mathematics and science. Some examples include:

• Electrical Engineering: Imaginary numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Imaginary numbers are used to represent complex signals and analyze their frequency content.
• Quantum Mechanics: Imaginary numbers are used to represent wave functions and analyze the behavior of particles at the quantum level.

Final Thoughts

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In conclusion, writing $\sqrt{-175}$ in its simplest radical form requires a deep understanding of the concept of imaginary numbers and the ability to manipulate complex expressions. By following the steps outlined in this article, we can simplify the expression and obtain the final answer.

Additional Resources

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For further reading on the topic of imaginary numbers and their applications, we recommend the following resources:

• Wikipedia: Imaginary number
• MathWorld: Imaginary number
• Khan Academy: Imaginary numbers

References

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• "Imaginary Numbers" by Michael Artin
• "Complex Analysis" by Serge Lang
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

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Introduction

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In our previous article, we explored how to write $\sqrt{-175}$ in its simplest radical form. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the definition of the imaginary unit $i$?

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A: The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. This means that $i^2 = -1$.

Q: How do I simplify $\sqrt{-175}$?

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A: To simplify $\sqrt{-175}$, you need to break down the number $-175$ into its prime factors. This will help you identify any perfect squares that can be extracted from the expression. Then, you can express $\sqrt{-175}$ in terms of $i$ and simplify the expression.

Q: What is the prime factorization of $-175$?

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A: The prime factorization of $-175$ is:

$$-175 = -1 \cdot 5^2 \cdot 7$$

Q: How do I express $\sqrt{-175}$ in terms of $i$?

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A: You can express $\sqrt{-175}$ in terms of $i$ by rewriting it as:

$$\sqrt{-175} = \sqrt{-1 \cdot 5^2 \cdot 7}$$

Q: What is the final simplified form of $\sqrt{-175}$?

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A: The final simplified form of $\sqrt{-175}$ is:

$$\sqrt{-175} = i \cdot 5 \cdot \sqrt{7}$$

Q: What are some real-world applications of imaginary numbers?

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A: Imaginary numbers have numerous applications in mathematics and science, including:

• Electrical Engineering: Imaginary numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Imaginary numbers are used to represent complex signals and analyze their frequency content.
• Quantum Mechanics: Imaginary numbers are used to represent wave functions and analyze the behavior of particles at the quantum level.

Q: What are some common mistakes to avoid when working with imaginary numbers?

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A: Some common mistakes to avoid when working with imaginary numbers include:

• Not using the correct definition of $i$: Make sure to use the correct definition of $i$ as the square root of $-1$.
• Not simplifying the expression correctly: Make sure to simplify the expression correctly by extracting any perfect squares.
• Not using the correct notation: Make sure to use the correct notation for imaginary numbers, including the use of $i$ and the square root symbol.

Q: What are some additional resources for learning more about imaginary numbers?

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A: Some additional resources for learning more about imaginary numbers include:

• Wikipedia: Imaginary number
• MathWorld: Imaginary number
• Khan Academy: Imaginary numbers

Q: How can I practice working with imaginary numbers?

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A: You can practice working with imaginary numbers by:

• Solving problems: Try solving problems that involve imaginary numbers, such as simplifying expressions and solving equations.
• Using online resources: Use online resources, such as Khan Academy and Mathway, to practice working with imaginary numbers.
• Working with a tutor: Work with a tutor who can provide guidance and support as you learn more about imaginary numbers.

Conclusion

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In this article, we have answered some frequently asked questions related to writing $\sqrt{-175}$ in its simplest radical form. We hope that this article has been helpful in clarifying any confusion and providing additional resources for learning more about imaginary numbers.