Write $\sqrt{-9}$ In I-form.A. $3i$ B. $6i$ C. $\sqrt{-3i}$ D. $3i^2$ E. $9i$

Introduction

In mathematics, the imaginary unit, denoted by $i$, is defined as the square root of $-1$. This concept is crucial in algebra, particularly when dealing with complex numbers. In this article, we will explore how to express negative numbers in the imaginary form, focusing on the specific case of $\sqrt{-9}$.

Understanding the Imaginary Unit

Before diving into the problem, it's essential to understand the properties of the imaginary unit. By definition, $i^2 = -1$. This means that any power of $i$ can be simplified using this property. For example, $i^3 = i^2 \cdot i = -1 \cdot i = -i$, and $i^4 = (i^2)^2 = (-1)^2 = 1$.

Expressing $\sqrt{-9}$ in i-form

To express $\sqrt{-9}$ in the imaginary form, we can start by rewriting $-9$ as $-1 \cdot 9$. This allows us to use the property of the imaginary unit to simplify the expression.

$\sqrt{-9} = \sqrt{-1 \cdot 9} = \sqrt{-1} \cdot \sqrt{9}$

Using the definition of the imaginary unit, we know that $\sqrt{-1} = i$. Therefore, we can rewrite the expression as:

$\sqrt{-9} = i \cdot \sqrt{9}$

Since $\sqrt{9} = 3$, we can simplify the expression further:

$\sqrt{-9} = i \cdot 3 = 3i$

Conclusion

In conclusion, the correct expression for $\sqrt{-9}$ in the imaginary form is $3i$. This result is obtained by using the properties of the imaginary unit and simplifying the expression step by step.

Comparison with Other Options

Let's compare our result with the other options provided:

• $6i$: This option is incorrect because it does not follow from the simplification of $\sqrt{-9}$.
• $\sqrt{-3i}$: This option is also incorrect because it involves a different expression that is not related to $\sqrt{-9}$.
• $3i^2$: This option is incorrect because it involves squaring the imaginary unit, which is not necessary to express $\sqrt{-9}$.
• $9i$: This option is incorrect because it does not follow from the simplification of $\sqrt{-9}$.

Final Answer

Introduction

In our previous article, we explored how to express negative numbers in the imaginary form, focusing on the specific case of $\sqrt{-9}$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$, is defined as the square root of $-1$. This concept is crucial in algebra, particularly when dealing with complex numbers.

Q: How do you simplify powers of the imaginary unit?

A: To simplify powers of the imaginary unit, you can use the property $i^2 = -1$. For example, $i^3 = i^2 \cdot i = -1 \cdot i = -i$, and $i^4 = (i^2)^2 = (-1)^2 = 1$.

Q: How do you express $\sqrt{-9}$ in the imaginary form?

A: To express $\sqrt{-9}$ in the imaginary form, you can start by rewriting $-9$ as $-1 \cdot 9$. This allows you to use the property of the imaginary unit to simplify the expression.

$\sqrt{-9} = \sqrt{-1 \cdot 9} = \sqrt{-1} \cdot \sqrt{9}$

Using the definition of the imaginary unit, you know that $\sqrt{-1} = i$. Therefore, you can rewrite the expression as:

$\sqrt{-9} = i \cdot \sqrt{9}$

Since $\sqrt{9} = 3$, you can simplify the expression further:

$\sqrt{-9} = i \cdot 3 = 3i$

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

A: $i$ and $-i$ are two distinct imaginary units. While $i$ is defined as the square root of $-1$, $-i$ is defined as the negative of the square root of $-1$. In other words, $i^2 = -1$ and $(-i)^2 = -1$.

Q: Can you express any negative number in the imaginary form?

A: Yes, you can express any negative number in the imaginary form. For example, $\sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{-1} \cdot \sqrt{16} = i \cdot 4 = 4i$.

Q: How do you express a negative number with a variable in the imaginary form?

A: To express a negative number with a variable in the imaginary form, you can use the same approach as before. For example, $\sqrt{-x^2} = \sqrt{-1 \cdot x^2} = \sqrt{-1} \cdot \sqrt{x^2} = i \cdot x = ix$.

Q: What are some common mistakes to avoid when expressing negative numbers in the imaginary form?

A: Some common mistakes to avoid when expressing negative numbers in the imaginary form include:

• Squaring the imaginary unit unnecessarily
• Not using the property of the imaginary unit to simplify the expression
• Not rewriting the negative number as a product of $-1$ and a positive number
• Not using the correct definition of the imaginary unit

Conclusion

In conclusion, expressing negative numbers in the imaginary form requires a clear understanding of the properties of the imaginary unit and the ability to simplify expressions using these properties. By following the steps outlined in this article, you can confidently express negative numbers in the imaginary form and avoid common mistakes.