Expressing i^123 - 4i^9 - 4i in Cartesian Form: A Simple Guide

Welcome to understanding powers of i, where we'll explore the fascinating patterns in complex numbers.Let's start by examining the fundamental pattern of powers of i.When we raise i to different powers, we get a repeating pattern every four steps.This pattern forms a circular rotation in the complex plane, moving counterclockwise.Each power of i represents a quarter turn counterclockwise, completing a full rotation every fourth power.Now, let's see how we can use this pattern to find larger powers of i, like i to the power of 123.We can divide 123 by 4, getting 30 with a remainder of 3.This means i to the 123 equals i to the power of 4 times 30, plus 3.We can rewrite this as i to the 4th power, raised to the 30th power, times i cubed.Since i to the 4th equals 1, and i cubed equals negative i...We can simplify this to negative i.Now that we understand how to handle large powers of i, let's move on to our next topic.For our second term negative four i to the ninth power, we'll use the division method to simplify.When we divide nine by four, we get a quotient of two and a remainder of one.This means i to the ninth can be written as i to the power of four times two, plus one.Since i to the fourth equals one, we can simplify this to one squared times i.Now we can multiply negative four by i to get negative four i.This gives us our three terms: negative i from the previous step, negative four i from this simplification, and negative four i which was already simplified.Now we'll combine all our terms with i to get our final answer.Each term came from a different part of our original expression.To combine like terms, we'll follow a systematic process.First, we identify all terms containing i. In this case, all three terms have i.Next, we add the coefficients: negative one, negative four, and negative four.Finally, we write our result with i.Our final answer, negative nine i, is now in Cartesian form, with a single real coefficient multiplied by i.We've successfully simplified our complex expression by understanding powers of i, simplifying each term, and combining like terms.Thanks for learning about complex numbers with Spark.E!