Which Expression Is Equivalent To { J \times J \times J \times J \times J \times J \times J \times J \times J \times J \times J \times J \times J $}$?A. { J^{13} $}$ B. { 13j $}$ C. { 1^{13} $}$ D. [$

Introduction

Imaginary numbers are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to extend the real number system to the complex number system, which is essential in solving equations and representing periodic phenomena. In this article, we will explore the properties of imaginary numbers, specifically the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We will examine the given options and determine which one is equivalent to this expression.

Imaginary Numbers and Their Properties

Imaginary numbers are defined as the square root of -1, denoted by $j$. This means that $j^2 = -1$. The properties of imaginary numbers are as follows:

• $j^2 = -1$
• $j^3 = -j$
• $j^4 = 1$
• $j^5 = j$
• $j^6 = -1$
• $j^7 = -j$
• $j^8 = 1$
• ...

As we can see, the powers of $j$ repeat in a cycle of four: $j, -1, -j, 1$. This means that any power of $j$ can be reduced to one of these four values.

Evaluating the Expression

Now, let's evaluate the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We can simplify this expression by grouping the powers of $j$:

$j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j = j^{13}$

Using the properties of imaginary numbers, we know that $j^4 = 1$. Therefore, we can reduce the power of $j$ by dividing it by 4:

$j^{13} = (j^4)^3 \times j = 1^3 \times j = j$

So, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$.

Analyzing the Options

Now, let's analyze the given options:

A. $j^{13}$ B. $13j$ C. $1^{13}$ D. $j$

As we have shown, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$. Therefore, the correct answer is:

The Correct Answer is D. $j$

Conclusion

In this article, we have explored the properties of imaginary numbers and evaluated the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We have shown that this expression is equivalent to $j$. We have also analyzed the given options and determined that the correct answer is D. $j$.

Imaginary Numbers and Their Properties

Imaginary numbers are defined as the square root of -1, denoted by $j$. This means that $j^2 = -1$. The properties of imaginary numbers are as follows:

• $j^2 = -1$
• $j^3 = -j$
• $j^4 = 1$
• $j^5 = j$
• $j^6 = -1$
• $j^7 = -j$
• $j^8 = 1$
• ...

As we can see, the powers of $j$ repeat in a cycle of four: $j, -1, -j, 1$. This means that any power of $j$ can be reduced to one of these four values.

Evaluating the Expression

Now, let's evaluate the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We can simplify this expression by grouping the powers of $j$:

$j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j = j^{13}$

Using the properties of imaginary numbers, we know that $j^4 = 1$. Therefore, we can reduce the power of $j$ by dividing it by 4:

$j^{13} = (j^4)^3 \times j = 1^3 \times j = j$

So, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$.

Analyzing the Options

Now, let's analyze the given options:

A. $j^{13}$ B. $13j$ C. $1^{13}$ D. $j$

As we have shown, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$. Therefore, the correct answer is:

The Correct Answer is D. $j$

Conclusion

In this article, we have explored the properties of imaginary numbers and evaluated the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We have shown that this expression is equivalent to $j$. We have also analyzed the given options and determined that the correct answer is D. $j$.

Imaginary Numbers and Their Properties

Imaginary numbers are defined as the square root of -1, denoted by $j$. This means that $j^2 = -1$. The properties of imaginary numbers are as follows:

• $j^2 = -1$
• $j^3 = -j$
• $j^4 = 1$
• $j^5 = j$
• $j^6 = -1$
• $j^7 = -j$
• $j^8 = 1$
• ...

As we can see, the powers of $j$ repeat in a cycle of four: $j, -1, -j, 1$. This means that any power of $j$ can be reduced to one of these four values.

Evaluating the Expression

Now, let's evaluate the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We can simplify this expression by grouping the powers of $j$:

$j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j = j^{13}$

Using the properties of imaginary numbers, we know that $j^4 = 1$. Therefore, we can reduce the power of $j$ by dividing it by 4:

$j^{13} = (j^4)^3 \times j = 1^3 \times j = j$

So, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$.

Analyzing the Options

Now, let's analyze the given options:

A. $j^{13}$ B. $13j$ C. $1^{13}$ D. $j$

As we have shown, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$. Therefore, the correct answer is:

The Correct Answer is D. $j$

Conclusion

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Q&A: Understanding Imaginary Numbers

Q: What is an imaginary number?

A: An imaginary number is a complex number that can be expressed in the form $a + bj$, where $a$ and $b$ are real numbers and $j$ is the imaginary unit, which satisfies the equation $j^2 = -1$.

Q: What is the imaginary unit $j$?

A: The imaginary unit $j$ is a mathematical constant that is defined as the square root of -1. It is denoted by the symbol $j$ and is used to extend the real number system to the complex number system.

Q: How do you simplify powers of $j$?

A: To simplify powers of $j$, you can use the following properties:

• $j^2 = -1$
• $j^3 = -j$
• $j^4 = 1$
• $j^5 = j$
• $j^6 = -1$
• $j^7 = -j$
• $j^8 = 1$
• ...

As you can see, the powers of $j$ repeat in a cycle of four: $j, -1, -j, 1$. This means that any power of $j$ can be reduced to one of these four values.

Q: How do you evaluate expressions with multiple powers of $j$?

A: To evaluate expressions with multiple powers of $j$, you can group the powers of $j$ and simplify them using the properties of $j$. For example, consider the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$. We can simplify this expression by grouping the powers of $j$:

$j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j = j^{13}$

Using the properties of $j$, we know that $j^4 = 1$. Therefore, we can reduce the power of $j$ by dividing it by 4:

$j^{13} = (j^4)^3 \times j = 1^3 \times j = j$

So, the expression $j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j \times j$ is equivalent to $j$.

Q: What are some common mistakes to avoid when working with imaginary numbers?

A: Some common mistakes to avoid when working with imaginary numbers include:

• Not using the correct properties of $j$
• Not simplifying powers of $j$ correctly
• Not grouping powers of $j$ correctly
• Not using the correct notation for imaginary numbers

Q: How do you add and subtract imaginary numbers?

A: To add and subtract imaginary numbers, you can use the following rules:

• $a + bj + c + dj = (a + c) + (b + d)j$
• $a + bj - c - dj = (a - c) + (b - d)j$

Q: How do you multiply imaginary numbers?

A: To multiply imaginary numbers, you can use the following rule:

• $(a + bj)(c + dj) = (ac - bd) + (ad + bc)j$

Q: How do you divide imaginary numbers?

A: To divide imaginary numbers, you can use the following rule:

• $\frac{a + bj}{c + dj} = \frac{(ac + bd) + (bc - ad)j}{c^2 + d^2}$

Conclusion

In this article, we have explored the properties of imaginary numbers and answered some common questions about them. We have shown how to simplify powers of $j$, evaluate expressions with multiple powers of $j$, and add, subtract, multiply, and divide imaginary numbers. We hope that this article has been helpful in understanding imaginary numbers and how to work with them.