Problem 89 - Intermediate Algebra

Question

Simplify each expression. $$ i^{21} $$

Step-by-Step Solution

Answer

The simplified expression is \( i \).

Step 1: Introduction to the Problem

We need to simplify the expression \( i^{21} \). The imaginary unit \( i \) has powers that follow a cyclical pattern. Therefore, understanding this cycle will help us simplify the expression.

Step 2: Understanding Powers of i

The powers of \( i \) cycle every four powers: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). After every four powers, the pattern repeats: \( i^5 = i \), \( i^6 = -1 \), and so on.

Step 3: Find the Remainder

To simplify \( i^{21} \), find the remainder when 21 is divided by 4, since the powers of \( i \) cycle every 4 powers. Calculate: \( 21 \div 4 = 5 \) remainder \( 1 \). So, \( 21 \equiv 1 \mod 4 \).

Step 4: Apply the Cycle

Use the cycle identified in Step 2. Since \( 21 \equiv 1 \mod 4 \), \( i^{21} = i^1 \). From our cycle, we know \( i^1 = i \). Thus, \( i^{21} = i \).

Key Concepts

Powers of i

Complex numbers often involve the concept of the imaginary unit, denoted as \( i \). This becomes especially interesting when we begin to explore the powers of \( i \). By definition, \( i \) is the solution to the equation \( x^2 = -1 \). Consequently, powers of \( i \) exhibit a predictable pattern. Understanding this pattern is crucial as it cycles through every fourth power. Here's what you need to know:

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

After reaching \( i^4 \), the cycle starts again at \( i^5 = i \), repeating the pattern. This feature of powers of \( i \) makes complex calculations simpler. Knowing this cycle, we can express any power of \( i \) in terms of these first four powers.

Cyclical Pattern

The cyclical pattern in powers of \( i \) is a fundamental concept when dealing with complex numbers. This cycle repeats every four exponents, meaning that after every fourth power, it starts over. The significance of this pattern lies in its ability to simplify computations involving high powers of \( i \). To determine which power corresponds to a specific exponent, like \( i^{21} \), you need to divide the exponent by 4 and focus on the remainder. This tells you which step of the cycle it corresponds to, following this logic:

• If remainder is 0, it corresponds to \( i^4 \)
• If remainder is 1, it corresponds to \( i^1 \)
• If remainder is 2, it corresponds to \( i^2 \)
• If remainder is 3, it corresponds to \( i^3 \)

By determining the remainder of 21 divided by 4, which is 1, we find that \( i^{21} = i^1 = i \). This method streamlines the process, turning potentially lengthy calculations into straightforward results.

Imaginary Unit

The imaginary unit \( i \) is at the heart of complex numbers. Introduced to extend the real number system, \( i \) provides solutions to equations that have no real solutions, specifically the square root of negative numbers. It is defined by the property \( i^2 = -1 \). This feature distinguishes \( i \) as an imaginary number and forms the basis for its applications in complex arithmetic. The behavior of \( i \) differs from real numbers, so understanding its properties, like the cyclical power pattern, is essential for solving complex number problems. The creation of \( i \) expanded the number system to what we now know as the complex plane, allowing for both real and imaginary numbers. These numbers are represented as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( bi \) expresses the imaginary component. This innovation broadens the scope of arithmetic and calculus, making it possible to address and solve problems that were once unsolvable.