Simplify $i^{52}$.$i^{52} = $

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Introduction

Understanding the Properties of Complex Numbers

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. The imaginary part is represented by the letter $i$, where $i = \sqrt{-1}$. In this article, we will focus on simplifying the expression $i^{52}$, which involves understanding the properties of complex numbers and their powers.

Review of Complex Number Properties

Before we dive into simplifying $i^{52}$, let's review some key properties of complex numbers:

• Definition of $i$: $i = \sqrt{-1}$
• Powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$
• Periodicity of $i$: The powers of $i$ repeat every 4th power, i.e., $i^{n+4} = i^n$

Simplifying $i^{52}$

To simplify $i^{52}$, we can use the periodicity of $i$. Since the powers of $i$ repeat every 4th power, we can divide 52 by 4 to find the remainder:

52 ÷ 4 = 13 with a remainder of 0

This means that $i^{52}$ is equivalent to $i^{4 \times 13}$, which is equal to $i^0$.

Evaluating $i^0$

$i^0$ is equal to 1, since any non-zero number raised to the power of 0 is equal to 1.

Conclusion

In conclusion, $i^{52}$ can be simplified to $i^0$, which is equal to 1.

Additional Examples

Here are some additional examples of simplifying powers of $i$:

• $i^3$: $i^3 = i^2 \times i = -1 \times i = -i$
• $i^5$: $i^5 = i^4 \times i = 1 \times i = i$
• $i^7$: $i^7 = i^4 \times i^3 = 1 \times -i = -i$

Final Thoughts

Simplifying powers of $i$ requires an understanding of the properties of complex numbers and their periodicity. By using the periodicity of $i$, we can simplify expressions like $i^{52}$ to their simplest form.

Frequently Asked Questions

• What is the value of $i^0$?
  • $i^0$ is equal to 1.
• What is the value of $i^{52}$?
  • $i^{52}$ is equal to $i^0$, which is equal to 1.
• How do you simplify powers of $i$?
  • You can simplify powers of $i$ by using the periodicity of $i$ and dividing the exponent by 4 to find the remainder.

References

• Complex Numbers
• Powers of $i$

Related Articles

• Simplifying Complex Expressions
• Properties of Complex Numbers

Final Conclusion

In conclusion, simplifying $i^{52}$ requires an understanding of the properties of complex numbers and their periodicity. By using the periodicity of $i$, we can simplify expressions like $i^{52}$ to their simplest form.

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Q&A: Simplifying Powers of $i$

Frequently Asked Questions

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

A: $i^0$ is equal to 1.

Q: What is the value of $i^{52}$?

A: $i^{52}$ is equal to $i^0$, which is equal to 1.

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

A: You can simplify powers of $i$ by using the periodicity of $i$ and dividing the exponent by 4 to find the remainder.

Q: What is the periodicity of $i$?

A: The powers of $i$ repeat every 4th power, i.e., $i^{n+4} = i^n$.

Q: How do you evaluate $i^3$?

A: $i^3 = i^2 \times i = -1 \times i = -i$.

Q: How do you evaluate $i^5$?

A: $i^5 = i^4 \times i = 1 \times i = i$.

Q: How do you evaluate $i^7$?

A: $i^7 = i^4 \times i^3 = 1 \times -i = -i$.

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

A: $i^1$ is equal to $i$.

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

A: $i^2$ is equal to $-1$.

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

A: $i^3$ is equal to $-i$.

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

A: $i^4$ is equal to $1$.

Additional Questions and Answers

Q: Can you provide more examples of simplifying powers of $i$?

A: Yes, here are some additional examples:

• $i^6 = i^4 \times i^2 = 1 \times -1 = -1$
• $i^8 = i^4 \times i^4 = 1 \times 1 = 1$
• $i^{10} = i^4 \times i^6 = 1 \times -1 = -1$

Q: How do you simplify expressions with multiple powers of $i$?

A: You can simplify expressions with multiple powers of $i$ by using the periodicity of $i$ and dividing each exponent by 4 to find the remainder.

Q: Can you provide a step-by-step guide to simplifying powers of $i$?

A: Yes, here is a step-by-step guide:

1. Divide the exponent by 4 to find the remainder.
2. Use the periodicity of $i$ to simplify the expression.
3. Evaluate the expression using the simplified form.

Conclusion

Simplifying powers of $i$ requires an understanding of the properties of complex numbers and their periodicity. By using the periodicity of $i$, we can simplify expressions like $i^{52}$ to their simplest form. We hope this Q&A article has provided you with a better understanding of how to simplify powers of $i$.

Related Articles

• Simplifying Complex Expressions
• Properties of Complex Numbers

References

• Complex Numbers
• Powers of $i$

Final Thoughts

Simplifying powers of $i$ is an important concept in mathematics, and it requires a good understanding of the properties of complex numbers and their periodicity. By using the periodicity of $i$, we can simplify expressions like $i^{52}$ to their simplest form. We hope this Q&A article has provided you with a better understanding of how to simplify powers of $i$.