Problem 81 - Intermediate Algebra

Question

Find each power of \(i .\) See Example 6. $$ 0.81 . i^{8} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.81.

Step 1: Understanding the Problem

We need to find the value of the expression \(0.81 \, i^8\). First, we need to recognize that \(i\) represents the imaginary unit, where \(i^2 = -1\). Powers of \(i\) repeat every four cycles.

Step 2: Identify the Pattern for Powers of i

The powers of \(i\) cycle as follows: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Knowing this cycle, any higher power of \(i\) can be simplified by dividing the exponent by 4 and using the remainder to determine the equivalent power within the cycle.

Step 3: Simplify i^8

To find \(i^8\), we note that 8 divided by 4 leaves a remainder of 0. Thus \(i^8\) corresponds to the same power as \(i^4\), which is 1.

Step 4: Calculate the Expression

Substitute \(i^8 = 1\) back into the expression, so it becomes \(0.81 \, \times \, 1\). Therefore, the value of the expression is 0.81.

Key Concepts

Complex Numbers

Complex numbers are a crucial part of mathematics, especially in fields involving electrical engineering and physics. They take the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The real number \( a \) is the real part, and the coefficient \( b \) of \( i \) is the imaginary part.
Understanding complex numbers allows us to solve equations that have no real solutions, such as \( x^2 + 1 = 0 \). Here, the solution involves \( x = \pm i \), since \( i^2 = -1 \).

• The real part \( a \) contributes to the x-axis on the complex plane.
• The imaginary part \( bi \) contributes to the y-axis.

This framework makes complex numbers essential for tasks spanning signal processing, fluid dynamics, and more.

Cycle of Powers

The concept of the cycle of powers is fundamental when working with the imaginary unit. Powers of \( i \), the imaginary unit, follow a predictable pattern, which repeats every four. This cycle is:

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

To find higher powers of \( i \), divide the power by 4 and use the remainder to locate the equivalent power in the cycle. For instance, for \( i^8 \), since 8 divided by 4 leaves a remainder of 0, this corresponds to \( i^4 \), which is 1.
This cyclic nature simplifies calculations by reducing higher powers to one of these four values.

Imaginary Unit

The imaginary unit, represented by \( i \), is a mathematical artifact introduced to manage square roots of negative numbers. By definition, \( i \) satisfies the equation \( i^2 = -1 \). This simple relationship creates a new dimension in algebra, enabling complex numbers.
Functions and operations that use \( i \) assume key roles in advanced mathematics and are involved in key concepts such as Fourier Transforms and fractals.

• It allows expressing solutions to polynomials that aren't solvable in the real number system.
• \( i \) helps reshape complex problems into manageable parts through its cyclic powers.

This imaginative leap in number theory underpins much of what we see in modern computational methods and signal processing.

Exponents and Powers

Exponents and powers are critical tools in mathematics that deal with how many times a number, called the base, is multiplied by itself. The notation \( b^n \) means that \( b \), the base, is raised to the exponent \( n \).
When applied to the imaginary unit, knowing how to simplify these powers becomes crucial, as illustrated in the cycle of powers for \( i \).
In our problem, recognizing that \( i^8 \) simplifies using the cycle helps to streamline the calculation by breaking down complex exponentials to known results.

• Uses include calculating growth processes, decay phenomena, and iterative algorithms.
• Exponents handle both integer and fractional powers for diverse applications.

This concept extends beyond numbers to functions, providing powerful insights into rate changes and periodicity.