Problem 43 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ i^{57} $$

Step-by-Step Solution

Answer

The simplest form of \( i^{57} \) is \( i \).

Step 1: Understand the Power of Imaginary Unit

Recall that the imaginary unit is defined as \( i = \sqrt{-1} \). Hence, whenever we need to find the power of \( i \), we look at its cyclical nature: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \). This cycle repeats every four powers.

Step 2: Simplify the Exponent

Given the problem, \( i^{57} \), we need to find where it falls in the cycle. Since powers of \( i \) repeat every 4, divide 57 by 4 to find the remainder using the division process or modulo operation: \[ 57 \div 4 = 14 \, ext{remainder} \, 1 \] Thus, we can express the exponent as 57 = 4 \times 14 + 1, meaning \( i^{57} = i^1 \).

Step 3: Apply the Remainder to Power Cycle

With the remainder being 1, we refer back to the cycle identified in Step 1: - \( i^1 = i \) Thus, \( i^{57} = i \), since 57 falls equivalently at the first position in the cycle.

Key Concepts

Understanding Powers of i

Imaginary numbers are fascinating mathematical tools which introduce the idea of powers, specifically those of the imaginary unit, denoted as \(i\). The imaginary unit is defined as \(i = \sqrt{-1}\). This definition leads to a set of repetitive results as you raise \(i\) to successive powers. Make sure to use these predictable patterns to your advantage.

The powers of \(i\) follow a cyclical pattern over four steps:

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

Once you reach \(i^4\), the cycle repeats again starting from \(i^5 = i\), \(i^6 = -1\), and so on. Remembering and using this cycle can simplify computations in algebra, helping you solve problems involving powers of imaginary numbers efficiently.

Simplifying Expressions with Imaginary Numbers

When it comes to simplifying expressions involving powers of \(i\), make use of the cyclical nature of these powers. This can dramatically reduce the complexity of problems. Consider the power in question and then identify the remainder when divided by 4. This helps pinpoint your position within the cycle pattern.

Suppose the task is to simplify \(i^{57}\). To find the equivalent lower power of \(i\), divide 57 by 4:
\[ 57 \div 4 = 14 \, \text{remainder} \, 1 \]
This calculation tells us that \(i^{57}\) is equivalent to \(i^1\) because the remainder is 1, falling back to the first position in the power cycle of \(i\). This simplifying method is valuable when dealing with any high power of \(i\), allowing you to reduce it to one of four simple results: \(i\), \(-1\), \(-i\), or \(1\). This approach not only saves time but also minimizes errors.

Introduction to Complex Numbers

A complex number is composed of both a real part and an imaginary part. Typically represented in the form \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component. Complex numbers extend the concept of one-dimensional number lines to two-dimensional planes, offering solutions to otherwise unsolvable equations.

Complex numbers are incredibly useful in various fields such as engineering, physics, and computer science. They allow complex calculations in circuitry or digital signal processing without oversimplifying reality by sticking to only real numbers. Simplifying complex numbers often involves separating the real part from the imaginary part, allowing computations and manipulations that are more efficient and straightforward.

For example, simplifying expressions with complex numbers might involve operations like addition, subtraction, and multiplication, which follows specific rules to keep terms properly combined into real and imaginary components separately. Understanding how imaginary numbers fit into the broader system of complex numbers helps you solve more advanced mathematical problems effectively.