Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{22} $$

Step-by-Step Solution

Answer

The simplified result of \(i^{22}\) is \(-1\).

Step 1: Understand the properties of i

The imaginary unit, represented by \(i\), satisfies the equation \(i^2 = -1\). Therefore, powers of \(i\) repeat every four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This sequence repeats for higher powers of \(i\).

Step 2: Determine the remainder when dividing the exponent by 4

To find \(i^{22}\), we first divide \(22\) by \(4\) to find the remainder:\(22 \div 4 = 5\) with a remainder of \(2\).

Step 3: Use the remainder to determine the power of i

Since the remainder is \(2\), \(i^{22}\) is equivalent to \(i^2\).

Step 4: Simplify using the known value for i^2

From Step 1, we know \(i^2 = -1\). Therefore, \(i^{22} = -1\).

Key Concepts

Imaginary Unit

The term "imaginary unit," denoted as \(i\), is a fundamental concept in complex numbers. It's unique because it's used to express the square root of \(-1\), something that isn't possible with real numbers. It might sound a bit abstract at first, but \(i\) is what allows us to deal with equations that involve negative square roots. For example, instead of saying \(\sqrt{-1}\), we can simply use \(i\). This simplification is incredibly helpful in various mathematical applications, including engineering and physics. Understanding \(i\) is key for delving into the world of complex numbers.

Powers of i

When dealing with powers of \(i\), it's important to notice a repeating pattern. This sequence helps simplify calculations involving higher powers. Here’s how it works:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Once \(i^4 = 1\) is reached, the cycle repeats. If you're tasked with finding a high power of \(i\), like \(i^{22}\), it's helpful to determine the remainder of the exponent divided by 4. This tells you where you are in the cycle, allowing you to easily simplify the expression. In our example, \(22 \div 4\) leaves a remainder of 2, indicating that \(i^{22} = i^2 = -1\). It’s a handy trick that simplifies calculations significantly.

Simplifying Complex Numbers

Simplifying complex numbers, especially when involving powers of \(i\), can seem confusing at first. However, once you understand the cycle of \(i\), it becomes much easier. Given any power of \(i\), you can use division and remainders to find out its equivalent simpler form.

• Determine the exponent modulo 4 (divide the exponent by 4 and find the remainder).
• Use the remainder to identify the equivalent power of \(i\) from the repeating cycle.

For instance, to simplify \(i^{22}\), divide 22 by 4. The remainder is 2, so \(i^{22}\) simplifies to \(i^2\), which equals \(-1\). This approach saves time and reduces complexity in mathematical calculations, making complex numbers more approachable and less daunting.