Problem 39

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{8} $$

Step-by-Step Solution

Verified

Answer

The result of $ i^8 $ is 1.

1Step 1: Understanding the Powers of i

The complex unit imaginary number $i$ is defined such that $i^2 = -1$. To understand higher powers of $i$, observe the pattern: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then the cycle repeats every four powers.

2Step 2: Express i^8 in Terms of the Cycle

Using the cycle identified ($i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$), determine where $i^8$ fits within this pattern. Notice that since $8$ is a multiple of $4$, $i^8$ will cycle back to the same result as $i^4$.

3Step 3: Calculate the Result

From the pattern established, $i^4 = 1$. Therefore, $i^8$ also equals $1$ because it's aligned with $i^4$.

4Step 4: Simplifying the Expression

The simplified form of $i^8$ is simply $1$, as it falls at the same position in the cycle as $i^4$.

Key Concepts

Understanding Powers of the Imaginary UnitSimplifying Complex ExpressionsPatterns in Powers of i

Understanding Powers of the Imaginary Unit

The imaginary unit, denoted as $i$, is a fundamental concept in complex numbers. It is uniquely defined by the property $i^2 = -1$. This single property allows us to effortlessly determine higher powers of $i$. For instance, let's observe the sequential powers:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

From $i^4$ onward, the powers of $i$ start to form a predictable cycle. This cycle repeats every four powers. Every time you calculate a power of $i$, you can determine its value by finding its position in this cycle. This cyclical nature is extremely useful for simplifying complex expressions.

Simplifying Complex Expressions

Simplifying complex expressions often involves dealing with powers of the imaginary unit $i$. By recognizing the repeating pattern of the powers—$i$, $-1$, $-i$, 1—you can simplify expressions that involve any power of $i$ by identifying its place within the cycle.
For instance, if you're asked to simplify $i^8$, start by noting the position of 8 in relation to the cycle. Since 8 is a multiple of the cycle length (4), $i^8$ will correspond to $i^4$, which is 1. Thus, the expression $i^8$ simplifies directly to 1, illustrating the power of understanding these basic patterns.
Visualizing or writing out the cycle can save time and reduce errors in computations. Simplifying doesn't mean breaking down every part; rather, it means recognizing the setup that allows you to work smarter.

Patterns in Powers of i

Patterns are at the heart of understanding mathematical concepts, and the powers of $i$ provide a great example. Because $i^2 = -1$, any further powers can be seen as simply building upon these foundational few:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

It's crucial to note that the sequence repeats itself. Thus, every fourth power cycle leads back to $i^4 = 1$. This insightful pattern not only simplifies calculations but also enhances understanding.
Therefore, when confronted with any power of $i$, such as $i^{37}$, you can find the remainder of 37 divided by 4, which is 1 in this case, showing $i^{37} = i^1 = i$. Recognizing and using these predictable patterns in powers of $i$ can greatly simplify working with complex numbers.

Problem 39

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