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 ​

Introduction

Radical expressions are a fundamental concept in mathematics, and they can be simplified using various techniques. However, when dealing with negative numbers under the square root, things can get a bit more complicated. In this article, we will explore the concept of imaginary numbers and how to simplify radical expressions involving negative numbers.

What are Imaginary Numbers?

Imaginary numbers are a type of complex number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. This means that $i^2 = -1$.

Simplifying Radical Expressions

When dealing with radical expressions involving negative numbers, we can use the concept of imaginary numbers to simplify them. Let's consider the expression $\sqrt{-80}$. To simplify this expression, we can start by factoring the number under the square root.

$\sqrt{-80} = \sqrt{-1 \cdot 80} = \sqrt{-1} \cdot \sqrt{80}$

Now, we can use the fact that $\sqrt{-1} = i$ to simplify the expression.

$\sqrt{-80} = i \cdot \sqrt{80}$

Next, we can simplify the square root of $80$ by factoring it into prime factors.

$\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5}$

Now, we can substitute this simplified expression back into the original expression.

$\sqrt{-80} = i \cdot 4 \sqrt{5} = 4i \sqrt{5}$

Conclusion

In conclusion, the expression $\sqrt{-80}$ is equivalent to $4i \sqrt{5}$. This is because we can simplify the radical expression by factoring the number under the square root and using the concept of imaginary numbers.

Answer

The correct answer is:

• B. $-4 i \sqrt{5}$ is incorrect because the correct expression is $4i \sqrt{5}$, not $-4i \sqrt{5}$.
• A. $-4 \sqrt{5}$ is incorrect because the correct expression involves the imaginary unit $i$.
• D. $4 \sqrt{5}$ is incorrect because the correct expression involves the imaginary unit $i$.

Final Answer

Q&A: Simplifying Radical Expressions

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 5. An imaginary number, on the other hand, is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the imaginary unit $i$? 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 a radical expression involving a negative number? A: To simplify a radical expression involving a negative number, you can use the concept of imaginary numbers. For example, to simplify $\sqrt{-80}$, you can start by factoring the number under the square root.

Q: What is the correct way to simplify $\sqrt{-80}$? A: The correct way to simplify $\sqrt{-80}$ is to factor the number under the square root and use the concept of imaginary numbers. This gives us:

$\sqrt{-80} = \sqrt{-1 \cdot 80} = \sqrt{-1} \cdot \sqrt{80} = i \cdot \sqrt{80}$

Next, we can simplify the square root of $80$ by factoring it into prime factors.

$\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5}$

Now, we can substitute this simplified expression back into the original expression.

$\sqrt{-80} = i \cdot 4 \sqrt{5} = 4i \sqrt{5}$

Q: What is the correct answer for the expression $\sqrt{-80}$? A: The correct answer is $4i \sqrt{5}$.

Q: What are some common mistakes to avoid when simplifying radical expressions? A: Some common mistakes to avoid when simplifying radical expressions include:

• Not factoring the number under the square root
• Not using the concept of imaginary numbers when dealing with negative numbers
• Not simplifying the square root of a number that can be factored into prime factors

Q: How do I know if a radical expression can be simplified using imaginary numbers? A: A radical expression can be simplified using imaginary numbers if it involves a negative number under the square root. In this case, you can use the concept of imaginary numbers to simplify the expression.

Q: What are some real-world applications of simplifying radical expressions? A: Simplifying radical expressions has many real-world applications, including:

• Calculating distances and lengths in geometry and trigonometry
• Solving equations and inequalities in algebra and calculus
• Modeling real-world phenomena in physics and engineering

Conclusion

In conclusion, simplifying radical expressions involving negative numbers requires an understanding of imaginary numbers. By factoring the number under the square root and using the concept of imaginary numbers, you can simplify radical expressions and solve equations and inequalities. Remember to avoid common mistakes and use the correct techniques to simplify radical expressions.

Final Answer

The final answer is $\boxed{4i \sqrt{5}}$.