Problem 79 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{-6}$$

Step-by-Step Solution

Answer

The simplified form of \(i^{-6}\) is \(-1\).

Step 1: Understanding the Cycle of i

The imaginary unit, represented by \(i\), follows a repeating cycle when raised to successive powers. This cycle is: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After \(i^4\), the cycle repeats. Understanding this cycle is crucial to simplifying powers of \(i\).

Step 2: Finding the Cycle Equivalent

To simplify \(i^{-6}\), we need to express it in terms of a positive power. Since \(i^{-n} = \frac{1}{i^n}\), we convert \(i^{-6}\) to \(\frac{1}{i^6}\). Our goal is to find \(i^6\).

Step 3: Applying the Cycle

We know that the powers of \(i\) repeat every 4. Thus, \(i^6\) can be reduced using the cycle by finding the remainder of 6 divided by 4: \(6 \div 4 = 1 ext{ R } 2\). So, \(i^6 = i^2\).

Step 4: Simplifying \(i^2\)

From the cycle, \(i^2 = -1\). Thus, \(i^6 = -1\).

Step 5: Simplifying \(\frac{1}{i^6}\)

Therefore, \(\frac{1}{i^6} = \frac{1}{-1} = -1\). So, \(i^{-6} = -1\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the study of complex numbers. Denoted by the symbol \(i\), it is defined mathematically as the square root of \(-1\). In other words, \(i^2 = -1\). This allows us to extend the set of real numbers to include solutions to equations that don't have real solutions, like \(x^2 = -1\).
When working with \(i\), it’s helpful to conceptualize it beyond its mathematical definition. Think of \(i\) as a tool that helps you perform calculations in the domain of complex numbers, which includes both a real part and an imaginary part. The imaginary unit \(i\) is what makes these complex numbers truly unique.

Powers of i

Understanding the powers of \(i\) is central to working with complex numbers. Powering \(i\) results in a predictable pattern that repeats every four terms:

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

After \(i^4\), the powers of \(i\) will start the cycle over again with \(i^5 = i\), \(i^6 = -1\), and so on. Each position in this cycle corresponds to a specific power of \(i\), determining whether the result is \(i, -1, -i,\) or \(1\). This cyclic nature simplifies calculations significantly by eliminating the need to compute high powers directly.

Cycle of Powers

The cycle of powers of \(i\) is an essential pattern that simplifies handling large exponents. When faced with powers of \(i\) greater than four, identifying how many times the cycle repeats helps find the equivalent lower power easier to compute.
For instance, given a power like \(-6\) in \(i^{-6}\), first convert to a positive power by expressing it as \(\frac{1}{i^6}\). Next, use the cycle pattern by dividing the exponent by \(4\), the length of the cycle. The remainder determines which part of the cycle the power corresponds to:
Divide \(6\) by \(4\) and get a remainder of \(2\). This means \(i^6\) is equivalent to \(i^2\), which we know from the cycle is \(-1\). Once that is found, calculations become straightforward, and you can quickly establish that \(i^{-6} = -1\). By understanding the cycle, complex calculations become manageable and far less daunting.