Which Expression Is Equivalent To − 80 \sqrt{-80} − 80 ​ ?A. − 4 5 -4 \sqrt{5} − 4 5 ​ B. − 4 I 5 -4 I \sqrt{5} − 4 I 5 ​ C. 4 I 5 4 I \sqrt{5} 4 I 5 ​ D. 4 5 4 \sqrt{5} 4 5 ​

Understanding the Problem

When dealing with square roots of negative numbers, we are essentially working with imaginary numbers. The expression $\sqrt{-80}$ can be simplified by first factoring out the negative sign and then finding the square root of the resulting expression.

Simplifying the Expression

We can start by factoring out the negative sign from the expression $\sqrt{-80}$. This gives us $\sqrt{-1} \cdot \sqrt{80}$. We know that $\sqrt{-1}$ is equivalent to $i$, where $i$ is the imaginary unit.

Simplifying the Square Root of 80

Now, we need to simplify the square root of 80. We can break down 80 into its prime factors, which are $2^4 \cdot 5$. Taking the square root of this expression gives us $\sqrt{2^4 \cdot 5} = 2^2 \cdot \sqrt{5} = 4 \sqrt{5}$.

Combining the Results

Now that we have simplified the square root of 80, we can combine it with the imaginary unit $i$. This gives us $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$.

Comparing the Results to the Options

We have found that the expression $\sqrt{-80}$ is equivalent to $4i \sqrt{5}$. Let's compare this result to the options provided:

• A. $-4 \sqrt{5}$: This option is incorrect because it does not take into account the imaginary unit $i$.
• B. $-4i \sqrt{5}$: This option is incorrect because it has a negative sign in front of the imaginary unit $i$, whereas our result has a positive sign.
• C. $4i \sqrt{5}$: This option is correct because it matches our result exactly.
• D. $4 \sqrt{5}$: This option is incorrect because it does not take into account the imaginary unit $i$.

Conclusion

In conclusion, the expression $\sqrt{-80}$ is equivalent to $4i \sqrt{5}$. This result can be obtained by first factoring out the negative sign from the expression and then simplifying the square root of 80. The final result is a combination of the imaginary unit $i$ and the square root of 5.

Key Takeaways

• When dealing with square roots of negative numbers, we are working with imaginary numbers.
• The expression $\sqrt{-80}$ can be simplified by factoring out the negative sign and then finding the square root of the resulting expression.
• The square root of 80 can be simplified by breaking it down into its prime factors and then taking the square root of the resulting expression.
• The final result is a combination of the imaginary unit $i$ and the square root of 5.

Frequently Asked Questions

• What is the imaginary unit $i$? The imaginary unit $i$ is a mathematical concept that represents the square root of -1. It is often used to simplify expressions involving square roots of negative numbers.
• How do I simplify the square root of a negative number? To simplify the square root of a negative number, you can factor out the negative sign and then find the square root of the resulting expression. This will give you a combination of the imaginary unit $i$ and the square root of the positive number.

Additional Resources

• For more information on imaginary numbers, see the article on "Imaginary Numbers" on our website.
• For more information on simplifying square roots, see the article on "Simplifying Square Roots" on our website.
  

Understanding the Basics

Simplifying square roots of negative numbers can be a challenging task, but with the right approach, it can be done easily. In this article, we will provide answers to some of the most frequently asked questions about simplifying square roots of negative numbers.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that represents the square root of -1. It is often used to simplify expressions involving square roots of negative numbers.

Q: How do I simplify the square root of a negative number?

A: To simplify the square root of a negative number, you can factor out the negative sign and then find the square root of the resulting expression. This will give you a combination of the imaginary unit $i$ and the square root of the positive number.

Q: What is the difference between $i$ and $-i$?

A: The imaginary unit $i$ is a positive imaginary number, while $-i$ is a negative imaginary number. When simplifying square roots of negative numbers, you will often encounter both $i$ and $-i$.

Q: How do I know which one to use?

A: When simplifying square roots of negative numbers, you will often need to use both $i$ and $-i$. The choice of which one to use will depend on the specific problem you are working on.

Q: Can I simplify square roots of negative numbers using only real numbers?

A: No, you cannot simplify square roots of negative numbers using only real numbers. The imaginary unit $i$ is a necessary component of simplifying square roots of negative numbers.

Q: What is the relationship between $i$ and $-i$?

A: The imaginary unit $i$ and $-i$ are conjugates of each other. This means that they have the same magnitude, but opposite signs.

Q: How do I simplify expressions involving $i$ and $-i$?

A: When simplifying expressions involving $i$ and $-i$, you can use the following rules:

• $i^2 = -1$
• $(-i)^2 = -1$
• $i \cdot -i = -i \cdot i = 1$

Q: Can I use $i$ and $-i$ interchangeably?

A: No, you cannot use $i$ and $-i$ interchangeably. While they are conjugates of each other, they have different properties and should be used carefully.

Q: What are some common mistakes to avoid when simplifying square roots of negative numbers?

A: Some common mistakes to avoid when simplifying square roots of negative numbers include:

• Forgetting to factor out the negative sign
• Using only real numbers to simplify square roots of negative numbers
• Interchanging $i$ and $-i$ without careful consideration

Q: How can I practice simplifying square roots of negative numbers?

A: You can practice simplifying square roots of negative numbers by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some real-world applications of simplifying square roots of negative numbers?

A: Simplifying square roots of negative numbers has many real-world applications, including:

• Electrical engineering
• Signal processing
• Quantum mechanics

Conclusion

Simplifying square roots of negative numbers can be a challenging task, but with the right approach, it can be done easily. By understanding the basics of the imaginary unit $i$ and how to simplify expressions involving $i$ and $-i$, you can become proficient in simplifying square roots of negative numbers. Remember to practice regularly and avoid common mistakes to improve your skills.

Additional Resources

• For more information on imaginary numbers, see the article on "Imaginary Numbers" on our website.
• For more information on simplifying square roots, see the article on "Simplifying Square Roots" on our website.
• For practice exercises and examples, see the article on "Practice Exercises: Simplifying Square Roots of Negative Numbers" on our website.