Problem 82 - Intermediate Algebra

Question

Find each power of \(i .\) See Example 6. $$ i^{10} $$

Step-by-Step Solution

Answer

\(i^{10} = -1\).

Step 1: Understand the Cycle of Powers of i

The imaginary unit \( i \) has a cyclical pattern for its powers. Specifically, the powers of \( i \) cycle every four exponents: - \( i^1 = i \) - \( i^2 = -1 \) - \( i^3 = -i \) - \( i^4 = 1 \) This pattern repeats for every next group of four powers.

Step 2: Divide the Exponent by 4

To find \( i^{10} \), divide the exponent by 4 and find the remainder. The remainder will help us determine which power in the cycle it corresponds to. Perform the division: \[ 10 \div 4 = 2 ext{ remainder } 2 \]So, the remainder when 10 is divided by 4 is 2.

Step 3: Use the Remainder to Find the Power

Refer to the cycle of powers of \( i \) determined in Step 1. Since the remainder is 2, \( i^{10} \) corresponds to \( i^2 \).Thus, \( i^2 = -1 \).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by \( i \), is a mathematical concept used to extend the real number system. It is defined by the property \( i^2 = -1 \). This means when you square the imaginary unit, you get \(-1\). This notion allows us to develop complex numbers, where a number can be expressed as a sum of a real part and an imaginary part. This innovation makes it possible to solve equations that lack real solutions, such as \( x^2 + 1 = 0 \). By introducing \( i \), we can say the solutions are \( x = i \) and \( x = -i \), fundamentally changing the landscape of algebra and calculus.

Cyclical Pattern

A fascinating feature of the powers of \( i \) is their cyclical pattern. This cycle is easy to recognize and memorize:

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

The pattern repeats every four exponents. Hence, any higher power of \( i \) can be reduced to one of these four outcomes, simply by determining its position in this cycle. Understanding this cycle simplifies working with powers of \( i \), as you only need to determine the remainder when dividing the exponent by 4 to identify its corresponding value in the sequence.

Remainder

To find where a specific power of \( i \) fits within its four-value cycle, we use division to find the remainder. For example, if we want to find \( i^{10} \), dividing 10 by 4 gives a remainder of 2. This remainder tells us which power in the sequence of \( i^1, i^2, i^3, i^4 \) corresponds to \( i^{10} \). In this instance, a remainder of 2 aligns with \( i^2 \), indicating that \( i^{10} = i^2 = -1 \). This method effectively reduces potentially complex calculations to simple arithmetic, leveraging the cyclical nature of the powers of \( i \).

Imaginary Numbers

Imaginary numbers are a fundamental extension of the real numbers, allowing us to solve equations that don't have solutions within the real number system. Each imaginary number is a multiple of the imaginary unit \( i \). When combined with real numbers, they form complex numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers.
Imaginary numbers broaden the scope of mathematical solutions, enabling new insights and applications. They play a crucial role in various fields, including engineering, physics, and computer science, where complex numbers are used to describe phenomena such as electrical currents and signal processing.