What Is The Square Root Of $-16$?A. $-8i$ B. $-4i$ C. $4i$ D. $8i$

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Introduction

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. However, when dealing with negative numbers, the concept of square roots becomes more complex. In this article, we will explore the square root of $-16$ and provide a detailed explanation of the correct answer.

Understanding Square Roots

The square root of a number $a$ is denoted by $\sqrt{a}$ and is a value that, when multiplied by itself, gives $a$. For example, the square root of $16$ is $4$ because $4 \times 4 = 16$. However, when dealing with negative numbers, the square root is not a real number. Instead, it is a complex number that involves the imaginary unit $i$, where $i = \sqrt{-1}$.

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They are denoted by $a + bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary unit $i$ is defined as the square root of $-1$, which means that $i^2 = -1$. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.

Square Root of $-16$

To find the square root of $-16$, we need to find a complex number that, when multiplied by itself, gives $-16$. We can start by expressing $-16$ as a product of $-1$ and $16$. Since $i^2 = -1$, we can write $-16$ as $(-1) \times 16$. Now, we need to find a complex number that, when multiplied by itself, gives $(-1) \times 16$.

Solving for the Square Root

Let's assume that the square root of $-16$ is a complex number $a + bi$. When we multiply this complex number by itself, we get:

$(a + bi)^2 = a^2 + 2abi + b^2i^2$

Since $i^2 = -1$, we can simplify this expression to:

$a^2 + 2abi - b^2 = -16$

Now, we need to find the values of $a$ and $b$ that satisfy this equation. We can start by equating the real and imaginary parts of the equation.

Equating Real and Imaginary Parts

Equating the real parts, we get:

$a^2 - b^2 = -16$

Equating the imaginary parts, we get:

$2ab = 0$

Since $a$ and $b$ are not necessarily zero, we can conclude that $ab = 0$. This means that either $a = 0$ or $b = 0$.

Solving for $a$ and $b$

If $a = 0$, then we have:

$-b^2 = -16$

Solving for $b$, we get:

$b = \pm 4$

If $b = 0$, then we have:

$a^2 = -16$

This equation has no real solution for $a$. Therefore, we can conclude that $b = \pm 4$.

Conclusion

In conclusion, the square root of $-16$ is a complex number that can be expressed as $-4i$ or $4i$. Therefore, the correct answer is:

The final answer is: B. $-4i$

Additional Examples

To further illustrate the concept of square roots of negative numbers, let's consider a few additional examples.

• The square root of $-25$ is $5i$.
• The square root of $-36$ is $6i$.
• The square root of $-49$ is $7i$.

In each of these examples, we can see that the square root of a negative number is a complex number that involves the imaginary unit $i$.

Conclusion

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Q: What is the square root of $-1$?

A: The square root of $-1$ is the imaginary unit $i$, which is defined as the number that, when multiplied by itself, gives $-1$. In other words, $i^2 = -1$.

Q: How do you find the square root of a negative number?

A: To find the square root of a negative number, you need to use the imaginary unit $i$. The square root of a negative number $-a$ can be expressed as $\sqrt{-a} = \sqrt{a}i$, where $\sqrt{a}$ is the square root of the positive number $a$.

Q: What is the square root of $-16$?

A: The square root of $-16$ is $-4i$ or $4i$, since $(-4i)^2 = (-4)^2 \times i^2 = 16 \times (-1) = -16$ and $(4i)^2 = (4)^2 \times i^2 = 16 \times (-1) = -16$.

Q: What is the square root of $-25$?

A: The square root of $-25$ is $5i$, since $(5i)^2 = (5)^2 \times i^2 = 25 \times (-1) = -25$.

Q: What is the square root of $-36$?

A: The square root of $-36$ is $6i$, since $(6i)^2 = (6)^2 \times i^2 = 36 \times (-1) = -36$.

Q: What is the square root of $-49$?

A: The square root of $-49$ is $7i$, since $(7i)^2 = (7)^2 \times i^2 = 49 \times (-1) = -49$.

Q: Can you give an example of a negative number that does not have a real square root?

A: Yes, the number $-1$ does not have a real square root. However, it does have an imaginary square root, which is the imaginary unit $i$ itself.

Q: Can you give an example of a negative number that has a real square root?

A: No, there is no negative number that has a real square root. By definition, the square root of a negative number is a complex number that involves the imaginary unit $i$.

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

A: A real number is a number that can be expressed without using the imaginary unit $i$. A complex number, on the other hand, is a number that involves the imaginary unit $i$.

Q: Can you give an example of a complex number?

A: Yes, the number $3 + 4i$ is a complex number, since it involves the imaginary unit $i$.

Q: Can you give an example of a real number?

A: Yes, the number $5$ is a real number, since it does not involve the imaginary unit $i$.

Conclusion

In this article, we have answered some frequently asked questions about square roots of negative numbers. We have seen that the square root of a negative number is a complex number that involves the imaginary unit $i$. We have also provided examples to illustrate the concept and have discussed the difference between real and complex numbers.