Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{15} $$

Step-by-Step Solution

Answer

The simplified form of \(i^{15}\) is \(-i\).

Step 1: Understand the Pattern of Powers of i

The imaginary unit \(i\) has a cyclic pattern of powers. The important powers to remember are: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats. This means that every 4th power, the result is 1, then the cycle starts again.

Step 2: Find the Remainder of the Exponent

To simplify \(i^{15}\), we first find the remainder when 15 is divided by 4. This will tell us which power within the cycle we are dealing with. Compute \(15 \div 4\), which equals 3, with a remainder of 3. The remainder tells us which power of \(i\) we need: \(i^{15} = i^3\).

Step 3: Simplify the Expression

Since we determined that \(i^{15} = i^3\), refer back to our cycle: \(i^3 = -i\). Substitute this into our expression, which gives the simplified result: \(-i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is a fascinating concept that comes into play in complex numbers. The imaginary unit, often represented by the letter \(i\), is defined as the square root of -1. But what does this mean?

• The imaginary unit \(i\) allows us to extend our number system beyond real numbers.
• It helps us express numbers that cannot be represented on the traditional number line.

So, when you see \(i\), you are dealing with a value that satisfies \(i^2 = -1\).
This concept might seem abstract at first but it's incredibly useful in various fields such as engineering and physics.

Powers of i

Understanding the powers of the imaginary unit \(i\) is key in simplifying expressions involving \(i\). As you raise \(i\) to higher powers, a cyclic pattern emerges:

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

After \(i^4\), the pattern repeats.
This repetition is crucial because it tells us that we don’t need to compute each power of \(i\) individually - we can use this pattern to simplify expressions quickly.
For example, to find \(i^{15}\), you determine where 15 falls in this cycle by considering the remainder when 15 is divided by 4. Here, the remainder is 3, which means \(i^{15}\) simplifies to \(i^3\), and therefore, \(-i\).

Cyclic Pattern

The cyclic pattern in the powers of \(i\) is one of those mathematically beautiful concepts that simplifies dealing with complex numbers.

• This pattern occurs because complex multiplication 'rotates' the imaginary unit in a circle on the complex plane.
• The cycle length for \(i\) is 4. Every 4 steps, you return to the same point in the cycle.

Understanding this cyclic pattern can be likened to solving a puzzle by recognizing the repeating pieces.
When faced with higher powers of \(i\), you just need to find where the power fits in the cycle, much like finding your location on a circular track.This convenient method helps avoid tedious calculations and makes working with powers of \(i\) much quicker and more intuitive.