Problem 77 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{64}$$

Step-by-Step Solution

Answer

\(i^{64} = 1\).

Step 1: Recall the Powers of i

The unit imaginary number \( i \) follows a specific cycle as its exponent increases: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). This cycle repeats every four powers.

Step 2: Determine the Cycle Position

Find the position of \( i^{64} \) within its cycle. This can be done by finding the remainder of 64 divided by 4 (because the cycle length is 4): \( 64 \div 4 = 16 \) with a remainder of 0.

Step 3: Identify the Equivalent Power

Since the remainder is 0, \( i^{64} \) is equivalent to \( i^0 \), and \( i^0 = 1 \). Thus, \( i^{64} \) simplifies to 1.

Key Concepts

Imaginary Unit

When you first hear about imaginary numbers, it might seem a bit puzzling. But don't worry, it's easier than it sounds! The foundation of all complex numbers is the imaginary unit, often denoted as \( i \). By definition, the imaginary unit \( i \) is

• the square root of -1, meaning that \( i = \sqrt{-1} \).

This might seem odd because in real numbers, you can't have the square root of a negative number. However, introducing \( i \) allows for solving equations that include negative roots. This extension of numbers enriches mathematical capabilities, enabling calculations and formulations in engineering and physics.
In essence, \( i \) is not just a number, but a pivotal concept that helps in operating with negative roots in complex number calculations.

Powers of i

The powers of \( i \) follow a fascinating repeating pattern. When considering different powers like \( i^1 \), \( i^2 \), and so on, you will notice a cyclical nature:

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

After \( i^4 \), the sequence restarts. This means any power of \( i \) beyond the fourth will repeat these same values. Understanding this cycle makes working with powers of \( i \) much simpler because you can predict the outcome based on just knowing the exponent's remainder when divided by 4.
Using \( i \) in equations becomes manageable as this cyclicity means you can reduce complex powers to simpler ones that are easy to calculate. Recognizing this repeating series not only simplifies calculations but reveals the elegant structure in mathematical patterns.

Exponential Cycle of i

The exponential cycle of \( i \) is a key element to simplifying powers of \( i \). Since we know the cycle repeats every four powers—\( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \)—you simply need to determine where your specific power falls within these steps.

• Take the exponent and divide by 4 to find the remainder.
• The remainder will tell you where in the cycle the power corresponds.

For example, if you have \( i^{64} \), dividing 64 by 4 gives a remainder of 0, reflecting that \( i^{64} \equiv i^0 \). Since \( i^0 \) is by convention equal to 1, you know that \( i^{64} = 1 \).
This method is especially useful for computing high powers of \( i \), where an exact direct calculation would be cumbersome. By recognizing the periodic cycle, what seems complex becomes quite straightforward. It's a clever way to simplify seemingly daunting expressions involving \( i \).