Which Choices Are Equivalent To The Expression Below? Check All That Apply. − 4 \sqrt{-4} − 4 ​ A. I 4 I \sqrt{4} I 4 ​ B. − 4 -\sqrt{4} − 4 ​ C. 2 I 2i 2 I D. − 2 -2 − 2

Introduction

In mathematics, complex expressions can be simplified using various techniques. One such technique is to express the square root of a negative number in terms of the imaginary unit, denoted by $i$. In this article, we will explore the simplification of the expression $\sqrt{-4}$ and determine which of the given choices are equivalent to it.

Understanding the Imaginary Unit

Before we dive into the simplification of the expression, let's briefly review the concept of the imaginary unit. The imaginary unit, denoted by $i$, is defined as the square root of $-1$. In other words, $i = \sqrt{-1}$. This concept is essential in simplifying complex expressions involving square roots of negative numbers.

Simplifying the Expression

Now, let's simplify the expression $\sqrt{-4}$. To do this, we can rewrite $-4$ as $-1 \times 4$. Using the property of square roots, we can rewrite the expression as:

$$\sqrt{-4} = \sqrt{-1 \times 4} = \sqrt{-1} \times \sqrt{4} = i \sqrt{4}$$

Evaluating the Choices

Now that we have simplified the expression, let's evaluate the given choices:

A. $i \sqrt{4}$

As we have just simplified, this choice is equivalent to the expression $\sqrt{-4}$.

B. $-\sqrt{4}$

This choice is not equivalent to the expression $\sqrt{-4}$ because it does not involve the imaginary unit $i$.

C. $2i$

This choice is not equivalent to the expression $\sqrt{-4}$ because it does not involve the square root of $4$.

D. $-2$

This choice is not equivalent to the expression $\sqrt{-4}$ because it does not involve the imaginary unit $i$.

Conclusion

In conclusion, the only choice that is equivalent to the expression $\sqrt{-4}$ is $i \sqrt{4}$. This is because it involves the imaginary unit $i$ and the square root of $4$, which are the key components of the simplified expression.

Additional Examples

To further illustrate the concept of simplifying complex expressions, let's consider a few additional examples:

Example 1

Simplify the expression $\sqrt{-9}$.

Using the same technique as before, we can rewrite $-9$ as $-1 \times 9$. Then, we can simplify the expression as:

$$\sqrt{-9} = \sqrt{-1 \times 9} = \sqrt{-1} \times \sqrt{9} = 3i$$

Example 2

Simplify the expression $\sqrt{-16}$.

Using the same technique as before, we can rewrite $-16$ as $-1 \times 16$. Then, we can simplify the expression as:

$$\sqrt{-16} = \sqrt{-1 \times 16} = \sqrt{-1} \times \sqrt{16} = 4i$$

Conclusion

In conclusion, simplifying complex expressions involving square roots of negative numbers requires a clear understanding of the imaginary unit and its properties. By using the property of square roots and the definition of the imaginary unit, we can simplify such expressions and determine which choices are equivalent to them.

Key Takeaways

• The imaginary unit, denoted by $i$, is defined as the square root of $-1$.
• To simplify an expression involving the square root of a negative number, we can rewrite the negative number as $-1$ times a positive number.
• Using the property of square roots, we can rewrite the expression as the product of the square root of $-1$ and the square root of the positive number.
• The resulting expression will involve the imaginary unit $i$ and the square root of the positive number.

Final Thoughts

Q: What is the imaginary unit, and how is it defined?

A: The imaginary unit, denoted by $i$, is defined as the square root of $-1$. In other words, $i = \sqrt{-1}$.

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

A: To simplify an expression involving the square root of a negative number, you can rewrite the negative number as $-1$ times a positive number. Then, using the property of square roots, you can rewrite the expression as the product of the square root of $-1$ and the square root of the positive number.

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

A: $i$ and $-i$ are conjugates of each other. In other words, $i$ is the square root of $-1$, and $-i$ is the negative of the square root of $-1$. When you multiply $i$ and $-i$, you get $-1$.

Q: Can I simplify an expression involving the square root of a negative number by just taking the square root of the absolute value of the number?

A: No, you cannot simplify an expression involving the square root of a negative number by just taking the square root of the absolute value of the number. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the absolute value of the number.

Q: How do I simplify an expression involving the square root of a negative number with a variable?

A: To simplify an expression involving the square root of a negative number with a variable, you can follow the same steps as before. However, you will need to use the variable in place of the number. For example, if you have the expression $\sqrt{-x}$, you can rewrite it as $\sqrt{-1} \times \sqrt{x}$.

Q: Can I simplify an expression involving the square root of a negative number by just using the absolute value of the number?

A: No, you cannot simplify an expression involving the square root of a negative number by just using the absolute value of the number. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the absolute value of the number.

Q: How do I know when to use the imaginary unit $i$?

A: You should use the imaginary unit $i$ whenever you encounter a square root of a negative number. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the absolute value of the number.

Q: Can I simplify an expression involving the square root of a negative number by just using the variable?

A: No, you cannot simplify an expression involving the square root of a negative number by just using the variable. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the variable.

Q: How do I simplify an expression involving the square root of a negative number with a fraction?

A: To simplify an expression involving the square root of a negative number with a fraction, you can follow the same steps as before. However, you will need to use the fraction in place of the number. For example, if you have the expression $\sqrt{-\frac{1}{2}}$, you can rewrite it as $\sqrt{-1} \times \sqrt{\frac{1}{2}}$.

Q: Can I simplify an expression involving the square root of a negative number by just using the fraction?

A: No, you cannot simplify an expression involving the square root of a negative number by just using the fraction. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the fraction.

Q: How do I know when to use the conjugate of a complex number?

A: You should use the conjugate of a complex number whenever you encounter a complex number in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Q: Can I simplify an expression involving the square root of a negative number by just using the conjugate of a complex number?

A: No, you cannot simplify an expression involving the square root of a negative number by just using the conjugate of a complex number. This is because the square root of a negative number involves the imaginary unit $i$, which is not present in the conjugate of a complex number.

Conclusion

In conclusion, simplifying complex expressions involving square roots of negative numbers requires a clear understanding of the imaginary unit and its properties. By following the steps outlined in this article, you can simplify complex expressions and determine which choices are equivalent to them.