Problem 35 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ 3 i+2 i^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \( i \).

Step 1: Simplify powers of i

We know the powers of the imaginary unit, i, follow a pattern: \( i = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then the pattern repeats. Here, we have \( i^3 = -i \).

Step 2: Substitute and simplify

Substitute \( i^3 \) with \( -i \) in the expression \( 3i + 2i^3 \), yielding \( 3i + 2(-i) \).

Step 3: Combine like terms

Combine the terms: \( 3i + 2(-i) = 3i - 2i = i \).

Key Concepts

Imaginary Unit

The concept of the imaginary unit, commonly denoted as \( i \), is a fundamental idea in complex numbers. Originally, mathematicians faced challenges when trying to find the square root of negative numbers. To solve this, they introduced \( i \), a number which satisfies the equation \( i^2 = -1 \). This means:\
\[\sqrt{-1} = i\]
With this definition, you're able to handle operations involving square roots of negative numbers smoothly.

• It allows for the extension of real numbers to complex numbers, which are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.
• This concept turns complex numbers into a powerful tool in various fields such as engineering, physics, and computer science.

Powers of i

Understanding the powers of the imaginary unit \( i \) is essential when working with complex numbers. The powers of \( i \) follow a cyclical pattern:

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

After \( i^4 \), the pattern repeats itself. So \( i^5 = i \), \( i^6 = -1 \), and so on.

To determine any power of \( i \), divide the exponent by 4 and use the remainder to find where it fits in the cycle:

• If the remainder is 0, the power of \( i \) is 1.
• If the remainder is 1, the power of \( i \) is \( i \).
• If the remainder is 2, the power of \( i \) is -1.
• If the remainder is 3, the power of \( i \) is -i.

Simplifying Expressions

Simplifying expressions involving complex numbers often revolves around handling the imaginary unit and its powers correctly. Here's a simple formula to consider: when given an expression like \( 3i + 2i^3 \), you should follow these steps to simplify:

• Identify and substitute powers of \( i \): Recognize that \( i^3 = -i \) from the pattern discussed previously.
• Replace \( i^3 \) with \(-i\) to get \( 3i + 2(-i) \).
• Combine like terms: \( 3i + 2(-i) = 3i - 2i = i \).

Utilizing these steps ensures you consistently simplify complex expressions accurately and efficiently, avoiding common pitfalls when dealing with imaginary numbers.