Problem 74 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{8}$$

Step-by-Step Solution

Answer

The simplified form of \(i^8\) is 1.

Step 1: Identify the Cycle of Powers of i

To understand how the powers of the imaginary unit \(i\) cycle through values, we list the initial powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). Notice that this sequence repeats every 4 powers. Therefore, any power of \(i\) can be simplified by finding its equivalent power within this cycle.

Step 2: Determine the Equivalent Power Within the Cycle

To find the equivalent power of \(i^8\) in the cycle, determine \(8 \mod 4\). Since there is no remainder when 8 is divided by 4, \(8 \mod 4 = 0\). This means \(i^8\) corresponds to \(i^4\) in the cycle.

Step 3: Simplify Using the Cycle

From Step 1, we know that \(i^4 = 1\). Thus, by the cyclical pattern, \(i^8 = 1\). Therefore, the power of \(i^8\) simplifies to 1.

Key Concepts

Powers of i

Imaginary numbers, notably represented by the symbol \(i\), constitute one of the foundations of complex number theory. The symbol \(i\) stands for the square root of -1. This is not a number that can be placed on a traditional number line. Instead, it's a crucial building block in complex mathematics. Understanding the powers of \(i\) is essential to simplify calculations involving imaginary numbers.

To explore the powers of \(i\), begin by observing how the powers of \(i\) evolve:

• \(i^1 = i\)
• \(i^2 = -1\) (since \(i \times i = i^2 = -1\))
• \(i^3 = -i\) (by continuing \(i^2 \times i = -1 \times i = -i\))
• \(i^4 = 1\) (because \(i^3 \times i = -i \times i = 1\))

Observe how this cycle repeats every four terms. By uncovering these core cycles, we simplify the use of \(i\) in complex arithmetic.

Cycling pattern of i

The cycling pattern of the powers of \(i\) is the key to simplifying larger expressions that involve imaginary numbers. After every four powers, the pattern repeats itself. This insight is immensely helpful in simplifying or calculating higher powers of \(i\).

To comprehend the cycle:

• The cycle of the powers of \(i\) is \(i, -1, -i, 1\).
• This means that every power of \(i\) can be condensed to one of these four key results: \(i\), \(-1\), \(-i\), or \(1\).

To find where a higher power, such as \(i^8\), lands in the cycle, use the modulo operation. Specifically, compute \(8 \mod 4\), which gives a remainder of 0—indicating that \(i^8\) aligns with \(i^4\) in the cycle and simplifies to 1. The modulo operation helps decode powers of \(i\) into simpler forms efficiently.

Simplifying complex numbers

Complex numbers combine real numbers with imaginary numbers to form a number of the form \(a + bi\), where \(a\) is the real component and \(b\) is the coefficient of the imaginary component \(i\). Simplifying complex numbers often involves using the cyclical nature of the powers of \(i\) to streamline expressions.

For example, if you encounter a term like \(i^8\) in a complex number expression, recognize it simplifies to 1 based on the cycling pattern of \(i\). This turns potentially complicated expressions into much simpler forms.

• If you're given an expression \(a + bi^8\), simplify \(bi^8\) to \(b\), resulting in \(a + b\).

This approach makes handling equations with complex numbers less daunting. By relying on the predictable nature of the powers of \(i\), complicated expressions become easy to manage, providing a clearer path to understanding and solving complex equations.