Problem 79 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-1\).

Step 1: Understand the Complex Number

Recognize that the imaginary unit is defined as \(i = \sqrt{-1}\). Hence, its powers repeat in a cycle every four powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats.

Step 2: Calculate using the Cycle

You need to simplify \(i^{-6}\). Start by considering the properties of \(i\). Each cycle is four, so divide the exponent by 4. Calculate as follows: \(-6 \div 4 = -1.5\). The remainder is \(-2\) since \(-1.5\) can be broken into full quarters such that \(-1.5 = -2 + 0.5\). As a result, \(i^{-6} = i^{4\times(-2) + 2} = (i^4)^{-2} \times i^2\).

Step 3: Apply the Cycle

Using the cycle, \(i^4 = 1\) so \((i^4)^{-2} = 1^{-2} = 1\). Consequently, \(i^{-6} = 1 \times i^2\).

Step 4: Simplify using Power

According to the cycle, \(i^2 = -1\), therefore simplify: \(1 \times i^2 = 1 \times (-1) = -1\). Thus, \(i^{-6} = -1\).

Key Concepts

Imaginary Unit

The imaginary unit, often denoted by the letter \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\). This concept might initially seem abstract since no real number squared equals \(-1\). However, in the realm of complex numbers, \(i\) allows for a broader spectrum of mathematical operations and applications.

Understanding the imaginary unit is crucial as it forms the building block for all other operations involving complex numbers. The introduction of \(i\) allows mathematicians to tackle problems related to square roots of negative numbers, which are impossible to solve within the set of real numbers.

In practical terms, any expression involving \(i\) constitutes a complex number. A complex number can be written as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, and \(bi\) denotes the imaginary part.

Powers of i

Powers of the imaginary unit \(i\) follow a unique cyclical pattern. This cyclical nature greatly simplifies calculations involving \(i\) raised to various powers. Understanding this pattern is key when solving expressions like \(i^{-6}\) as per the original exercise.

The sequence of powers of \(i\) can be remembered as a cycle:

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

After the fourth power, the cycle repeats itself:

\(i^5 = i\)

and so on.

Due to this pattern, any power of \(i\) can be simplified by determining its equivalent within this cycle.

For large, positive or negative powers such as \(i^{-6}\), using integer division of the exponent by 4 helps to locate its position within the cycle. The remainder helps identify the equivalent power of \(i\). This approach massively simplifies what otherwise might appear complex at first glance, reducing calculations to just a few basic operations.

Cyclic Properties of Complex Numbers

Cyclic properties significantly influence how we perform calculations with complex numbers. The cycle formed by the powers of \(i\) is a quintessential example of these properties and aids in simplifying complex numerical expressions.

To break it down: every time we multiply by \(i\), we move to the next step in the cycle. Thus, understanding the cycle allows us to predict the result of any power of \(i\) easily. This cyclic behavior also extends into more advanced operations, paving the way for efficient computation within the complex number system.

In the context of the original exercise \(i^{-6}\), recognizing and utilizing these cyclic properties allows you to manage numbers with large or negative exponents effortlessly. By employing the cycle, we can reduce any exponent - positive, zero, or negative - and predict exactly which segment of the repeating cycle it corresponds to, making calculations clearer and faster.

These cyclic properties don't just lighten computational load in mathematical theory but have practical applications too, such as simplifying complex electronic signals in engineering and other scientific computations.