Problem 39 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ i+i^{3}+i^{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(i\).

Step 1: Identify Powers of i

Recall the powers of the imaginary unit \(i\): - \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = -i\)- \(i^4 = 1\)These four results repeat cyclically. We'll use these to simplify the expression.

Step 2: Simplify Each Term

Replace each power of \(i\) in the expression with its corresponding value:- \(i^1 = i\)- \(i^3 = -i\)- \(i^5 = i \) since \(i^5 = i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i\)Thus, the expression simplifies to: \[ i + (-i) + i \]

Step 3: Combine Like Terms

Combine the terms: \[ i - i + i = i\]The negative and positive \(i\) cancel each other out, leaving just \(i\).

Key Concepts

Understanding the Powers of i

The concept of imaginary numbers might seem daunting at first. However, once you understand the powers of the basic imaginary unit \(i\), everything becomes clearer. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). From this definition, additional powers of \(i\) follow a repeating cycle of values:

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

Once we reach \(i^4\), the cycle restarts. This pattern is key when simplifying expressions containing powers of \(i\) because any power can be reduced to one of these four values.
For example, \(i^5\) is the same as \(i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i\). So understanding this cycle allows us to simplify powers of \(i\) easily.

Exploring Complex Numbers

Complex numbers are an extension of the real numbers and come into play when dealing with square roots of negative numbers. A complex number is written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
Here's what each part represents:

• \(a\): the real part of the complex number
• \(bi\): the imaginary part of the complex number

Imaginary numbers like \(i, -i\), and so forth, are specific examples of complex numbers where the real part is zero. In practice, complex numbers are used across various fields such as engineering and physics because they allow us to solve equations that no real number can solve, providing a more complete and satisfying framework for mathematics.
Whenever you encounter an expression like \(i + (-i) + i\), you're effectively dealing with complex numbers, albeit with some real parts being zero.

Techniques for Simplifying Expressions

Simplifying expressions involving imaginary numbers often boils down to reducing powers of \(i\) and combining like terms. Let's see how this works with the expression \(i + i^3 + i^5\). First, we use our knowledge of the powers of \(i\):

• \(i^1 = i\)
• \(i^3 = -i\)
• \(i^5 = i\)

Once we've substituted these in, the expression becomes \(i + (-i) + i\). Now, it's all about combining like terms. Here, \(i + (-i)\) simplifies to \(0\), as they cancel each other out.
You're left with \(i\), effectively simplifying the entire expression down to a single term. Keep in mind that when you're simplifying such expressions, following the cyclic pattern of powers of \(i\) and meticulously combining terms will lead you to the simplest form. This approach clears away confusion and makes the expression more manageable.