Problem 82 - Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i\) $$ i^{100} $$

Step-by-Step Solution

Answer

\(i^{100} = 1 + 0i\)

Step 1: Understand the Pattern of Powers of i

We start by understanding the cyclic nature of powers of the imaginary unit, \(i\). The fundamental powers are: \(\begin{align*}i^1 & = i ewline i^2 & = -1 ewline i^3 & = -i ewline i^4 & = 1\end{align*}\)These values repeat every four powers, forming a cycle.

Step 2: Determine the Remainder of 100 Divided by the Cycle Length

The powers of \(i\) repeat every 4. So, to simplify \(i^{100}\), we need to find the remainder of 100 when divided by 4:\[ 100 \div 4 = 25 \quad ext{with a remainder of } 0\]

Step 3: Use the Remainder to Simplify the Expression

Since \(100\mod 4 = 0\), this means \(i^{100} = i^{4k}\) for some integer \(k\). As \(i^4 = 1\), any power of \(i\) that is a multiple of 4 simplifies to 1. Thus, \(i^{100} = 1\).

Key Concepts

Imaginary Unit

The imaginary unit, often represented as \( i \), is a mathematical concept that allows us to extend the number system beyond real numbers. It is defined by the equation \( i^2 = -1 \). This is a unique attribute because no real number squared equals \(-1\). Thus, \( i \) becomes the foundation for complex numbers.

Complex numbers combine real numbers and multiples of \( i \), and they are usually written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The imaginary unit is crucial because it expands our ability to solve equations that would otherwise have no solution in the real numbers. For example, the quadratic equation \( x^2 + 1 = 0 \) has no real solutions, but it has solutions \( x = i \) and \( x = -i \) in the complex number system.

Understanding \( i \) is fundamental for anyone delving into complex numbers, as it sits at the heart of many complex operations and functions.

Powers of i

The powers of \( i \) follow a repetitive cycle. Understanding this cycle is key to simplifying expressions that involve powers of \( i \). Let's explore these powers:

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

This sequence repeats every four powers, creating a periodic pattern. Therefore, any power of \( i \) can be reduced to one of these four values by finding the remainder of the exponent when divided by 4. For instance, if you need to determine \( i^{100} \), you divide 100 by 4, which leaves a remainder of 0. This means \( i^{100} \) simplifies to \( i^0 \), which is equivalent to 1.

Embracing this cyclic nature greatly simplifies working with powers of \( i \), allowing you to convert even large exponents into manageable forms.

Simplifying Complex Expressions

Simplifying complex expressions often involves reducing powers of the imaginary unit \( i \) and combining like terms. Here's how you can tackle these expressions step by step:

• Identify the components of the complex expression. Look for real parts and imaginary parts, recognizing that any powers of \( i \) need special attention due to their cyclical nature.
• Reduce powers of \( i \) using the power cycle: \( i^1, i^2, i^3, i^4 \). For example, if an expression includes \( i^{100} \), simplify it to 1 since \( i^{100} = (i^4)^{25} \).
• Express the result in the standard form \( a + bi \), where \( a \) is the real component, and \( b \) is the coefficient of the imaginary part.

To illustrate, take \( i^{100} \):
1. Recognize the power of \( i \).2. Determine the remainder of 100 divided by 4, which is 0.3. Match the remainder to the cyclic pattern of \( i \), thus simplifying \( i^{100} \) to 1.

Following these steps helps to clean up expressions, making them easier to handle and ensuring they are expressed in a clear and concise manner.