What Is The Value Of The Expression I 0 × I 1 × I 2 × I 3 × I 4 I^0 \times I^1 \times I^2 \times I^3 \times I^4 I 0 × I 1 × I 2 × I 3 × I 4 ?A. 1 B. -1 C. I D. -i

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Introduction to Complex Numbers

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. A complex number 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, which satisfies the equation $i^2 = -1$. In this article, we will explore the value of the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$.

Properties of Imaginary Unit

The imaginary unit $i$ has several important properties that we need to understand in order to evaluate the given expression. One of the most important properties of $i$ is that it satisfies the equation $i^2 = -1$. This means that $i^2$ is equal to the negative of the number 1. We can also write $i^2$ as $-1$. Another important property of $i$ is that it is a root of the equation $x^2 + 1 = 0$. This means that $i$ is a solution to the equation $x^2 + 1 = 0$.

Evaluating Powers of $i$

To evaluate the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$, we need to understand how to evaluate powers of $i$. We can start by evaluating the powers of $i$ individually. We know that $i^2 = -1$, so we can write $i^4$ as $(i^2)^2 = (-1)^2 = 1$. Similarly, we can write $i^3$ as $i^2 \times i = -1 \times i = -i$. We can also write $i^1$ as simply $i$. Finally, we know that $i^0$ is equal to 1.

Evaluating the Expression

Now that we have evaluated the powers of $i$ individually, we can evaluate the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$. We can start by substituting the values we found earlier. We have $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. Substituting these values into the expression, we get:

$$i^0 \times i^1 \times i^2 \times i^3 \times i^4 = 1 \times i \times (-1) \times (-i) \times 1$$

Simplifying the Expression

Now that we have substituted the values we found earlier, we can simplify the expression. We can start by multiplying the terms together. We have:

$$1 \times i \times (-1) \times (-i) \times 1 = i \times (-1) \times (-i) \times 1$$

Using the Properties of $i$

We can use the properties of $i$ to simplify the expression further. We know that $i^2 = -1$, so we can write $-i \times i$ as $-i^2 = -(-1) = 1$. Substituting this value into the expression, we get:

$$i \times (-1) \times 1 \times 1 = -i$$

Conclusion

In this article, we evaluated the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$. We used the properties of the imaginary unit $i$ to simplify the expression and found that the value of the expression is $-i$.

Final Answer

The final answer to the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$ is $\boxed{-i}$.

Discussion

The expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$ is a classic example of how to use the properties of the imaginary unit $i$ to simplify complex expressions. The key to solving this problem is to understand the properties of $i$ and how to use them to simplify the expression. We hope that this article has provided a clear and concise explanation of how to evaluate this expression.

Related Topics

• Complex Numbers
• Imaginary Unit
• Properties of $i$
• Simplifying Expressions

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Imaginary Unit" by Wolfram MathWorld
• [3] "Properties of $i$" by Khan Academy

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Frequently Asked Questions

We have received many questions from readers about evaluating the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the value of $i^0$?

A: The value of $i^0$ is 1. This is because any number raised to the power of 0 is equal to 1.

Q: What is the value of $i^1$?

A: The value of $i^1$ is $i$. This is because $i$ is the imaginary unit, and it is defined as the square root of -1.

Q: What is the value of $i^2$?

A: The value of $i^2$ is -1. This is because $i^2$ is equal to the negative of the number 1.

Q: What is the value of $i^3$?

A: The value of $i^3$ is $-i$. This is because $i^3$ is equal to $i^2 \times i$, and we know that $i^2$ is equal to -1.

Q: What is the value of $i^4$?

A: The value of $i^4$ is 1. This is because $i^4$ is equal to $(i^2)^2$, and we know that $i^2$ is equal to -1.

Q: How do I evaluate the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$?

A: To evaluate the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$, you can start by substituting the values we found earlier. We have $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. Substituting these values into the expression, we get:

$$i^0 \times i^1 \times i^2 \times i^3 \times i^4 = 1 \times i \times (-1) \times (-i) \times 1$$

Q: How do I simplify the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$?

A: To simplify the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$, you can start by multiplying the terms together. We have:

$$1 \times i \times (-1) \times (-i) \times 1 = i \times (-1) \times (-i) \times 1$$

Q: What is the final answer to the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$?

A: The final answer to the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$ is $\boxed{-i}$.

Additional Resources

If you have any additional questions about evaluating the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$, you can find more information in the following resources:

• [1] "Complex Numbers" by Math Open Reference
• [2] "Imaginary Unit" by Wolfram MathWorld
• [3] "Properties of $i$" by Khan Academy

Conclusion

We hope that this article has provided a clear and concise explanation of how to evaluate the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$. If you have any additional questions or need further clarification, please don't hesitate to ask.

Final Answer

The final answer to the expression $i^0 \times i^1 \times i^2 \times i^3 \times i^4$ is $\boxed{-i}$.