Problem 86 - Intermediate Algebra

Question

Find each power of \(i .\) See Example 6. $$ i^{40} $$

Step-by-Step Solution

Answer

\( i^{40} = 1 \).

Step 1: Understand the Powers of i

The imaginary unit, denoted as \( i \), has a cyclic pattern when raised to successive powers. The cycle is as follows:\[ \begin{align*}i^1 &= i, \i^2 &= -1, \i^3 &= -i, \i^4 &= 1. \i^5 &= i \&\text{and the cycle repeats.} \end{align*}\]This means every four powers, the cycle resets.

Step 2: Determine the Cyclic Position for i^{40}

Since the powers of \( i \) repeat every 4, we need to find where 40 falls in the cycle. We do this by dividing the exponent 40 by 4:\[ 40 \div 4 = 10 \text{ remainder } 0 \].A remainder of 0 means \( i^{40} \) corresponds to \( i^4 \), which is the completion of 10 cycles.

Step 3: Evaluate i^4 Based on the Cycle

From the cyclic pattern of \( i \), we know that \( i^4 = 1 \). Therefore, any power of \( i \) that lands on a multiple of 4 is equal to 1.

Key Concepts

Powers of i

When dealing with imaginary numbers, understanding the powers of \( i \) is crucial. The imaginary unit \( i \), which represents the square root of -1, showcases unique properties when raised to successive powers. Specifically, \( i \) cycles through a predictable pattern every four powers. Here's how:

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

This means if you continue to raise \( i \) to higher powers, it keeps looping back through this set pattern. This repetition can be incredibly useful when solving problems, as it allows you to simplify expressions that might initially seem complex.
For instance, simplifying a high power like \( i^{40} \) involves recognizing that it will follow the same cycle, potentially reducing challenging calculations to quick observations.

Complex Numbers

Complex numbers extend the idea of real numbers by incorporating the imaginary unit \( i \). A complex number typically takes the form \( a + bi \), where \( a \) and \( b \) are real numbers. In this form:

• \( a \) is known as the real part
• \( b \) is the imaginary part

Both parts are necessary to fully represent the number on a two-dimensional plane, often referred to as the complex plane. On this plane, the horizontal axis represents the real part, while the vertical axis corresponds to the imaginary part.
Complex numbers are versatile in that they can represent values that are not possible with just real numbers alone. They are particularly important in fields such as engineering, physics, and applied mathematics.
When adding, subtracting, multiplying, or dividing complex numbers, understanding how \( i \) behaves under various operations is essential. This involves keeping in mind the cyclic nature of \( i \)'s powers.

Cyclic Pattern of i

The cyclic pattern of \( i \) is a central feature of its mathematical behavior. Repeatedly using \( i \) to compute complex operations becomes straightforward once you unravel the cycle:

• Since \( i^4 = 1 \), any power of \( i \) that is a multiple of 4 will equal 1.
• If a power is one more than a multiple of 4, it will equal \( i \).
• If two more, it equals \(-1\).
• If three more, it turns to \(-i\).

This cyclic nature allows you to quickly identify high powers of \( i \) by determining their position in the cycle. For example, calculating \( i^{40} \) is simplified by dividing the exponent by 4, which reveals a remainder of 0. Hence \( i^{40} \) aligns with \( i^4 \), equaling 1.
This example illustrates how understanding and utilizing the cyclic pattern minimizes computation and aids efficient problem-solving.