How to Simplify i to a Power
In mathematics, the imaginary unit “i” is a crucial component in the complex number system. Often, when dealing with complex numbers, we need to simplify expressions involving powers of “i”. This article aims to provide a step-by-step guide on how to simplify “i” to a power, making it easier to work with complex numbers in various mathematical contexts.
Understanding the Basics
Before diving into the simplification process, it is essential to understand the properties of the imaginary unit “i”. The following properties are fundamental in simplifying powers of “i”:
1. i^2 = -1
2. i^3 = -i
3. i^4 = 1
These properties hold true for all real numbers “n”. By using these properties, we can simplify expressions involving powers of “i”.
Step-by-Step Guide to Simplifying i to a Power
To simplify “i” to a power, follow these steps:
1. Identify the exponent “n” of “i”.
2. Divide “n” by 4 to find the quotient “q” and the remainder “r”.
3. Use the properties of “i” to express “i^n” in terms of “i^q” and “i^r”.
4. Simplify “i^r” using the properties of “i”.
5. Combine the simplified terms to obtain the final result.
Let’s go through an example to illustrate this process:
Example: Simplify i^15
1. Identify the exponent “n” of “i”: n = 15.
2. Divide “n” by 4: 15 ÷ 4 = 3 with a remainder of 3.
3. Express “i^15” in terms of “i^3” and “i^3”: i^15 = (i^4)^3 i^3.
4. Simplify “i^3” using the properties of “i”: i^3 = -i.
5. Combine the simplified terms: (i^4)^3 i^3 = 1^3 (-i) = -i.
Therefore, i^15 simplifies to -i.
Conclusion
Simplifying “i” to a power is an essential skill in working with complex numbers. By understanding the properties of the imaginary unit “i” and following the step-by-step guide provided in this article, you can simplify expressions involving powers of “i” with ease. This knowledge will enable you to tackle more complex problems in mathematics and other scientific fields.
