Problem 105 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ i^{6}+i^{8} $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Understanding Powers of i

First, let's remember the pattern of the powers of the imaginary unit, \(i\). We know that \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This pattern repeats every four powers.

Step 2: Evaluate i to the Sixth Power

To find \(i^6\), notice that \(i^6 = (i^4) imes (i^2)\). From the pattern established earlier, \(i^4 = 1\) and \(i^2 = -1\). Therefore, \(i^6 = 1 imes (-1) = -1\).

Step 3: Evaluate i to the Eighth Power

Similarly, to find \(i^8\), recognize that \(i^8 = (i^4) imes (i^4)\). As \(i^4 = 1\), this gives us \(i^8 = 1 imes 1 = 1\).

Step 4: Combine Results

Now add the results from both calculations, \(-1 + 1\). This simplifies to \(0\), which means the imaginary part is also \(+0i\). So, the expression \(i^6 + i^8\) simplifies to \(0\).

Key Concepts

Imaginary Unit

In mathematics, especially in complex number systems, the imaginary unit is a fundamental concept. The imaginary unit is denoted by \(i\), which is defined as the square root of \(-1\).
This means that \(i^2 = -1\), and it is the cornerstone of building complex numbers. Complex numbers take the form \(a + bi\), where \(a\) is called the real part, and \(bi\) is the imaginary part.
It’s important to note that while \(i\) itself isn’t a real number, expressions involving \(i\) can be evaluated to produce real results when appropriate.
The introduction of the imaginary unit allows for solutions to equations that have no real solutions, such as \(x^2 + 1 = 0\). Before \(i\) was defined, such equations had no solutions within the realm of real numbers.

Powers of i

When it comes to calculating powers of \(i\), there is a simple pattern that repeats every four powers. Understanding this pattern is crucial when dealing with complex number calculations.
Here is the cycle:

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

After reaching \(i^4 = 1\), the cycle begins again starting with \(i^5 = i\), \(i^6 = -1\), \(i^7 = -i\), and \(i^8 = 1\), and so on.
This repeating pattern is helpful because it allows us to simplify larger powers of \(i\) by finding the equivalent power within the first four terms. To find a specific power of \(i\), you can divide the exponent by 4 and use the remainder to identify the corresponding value in the cycle.

Complex Number Addition

The process of adding complex numbers involves summing their real parts and imaginary parts separately.
If we have two complex numbers, \(a + bi\) and \(c + di\), their sum is \((a+c) + (b+d)i\).
This method allows us to combine both parts smoothly. One thing to keep in mind is that we treat the imaginary unit \(i\) as a variable, thus combining coefficients directly applies to the imaginary terms just like the real ones.
In the context of our initial problem, \(i^6 + i^8\), once we simplify each part using the powers of \(i\) rule, we end up with real components that merely add or subtract depending on their signs, resulting in a simple zero, \(0 + 0i\).