Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Simplify each expression to \(i, 1,-i,\) or \(-1\) $$\frac{1}{i^{12}}$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Recall the powers of i

The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\). Its powers cycle every four terms: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats for higher powers.

Step 2: Find the remainder

To simplify \(\frac{1}{i^{12}}\), first determine \(i^{12}\) by finding the remainder when dividing 12 by 4, the length of \(i\)'s power cycle. Calculate \(12 \div 4\) which gives a quotient of 3 with a remainder of 0.

Step 3: Determine the equivalent power of i

Since the remainder is 0, \(i^{12}\) is equivalent to \(i^0\). By definition, any number raised to the power of 0, including \(i\), is 1. Therefore, \(i^{12} = 1\).

Step 4: Simplify the original expression

Substitute the value of \(i^{12}\) into the original expression: \(\frac{1}{i^{12}} = \frac{1}{1} = 1\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is foundational to understanding complex numbers. Invented to solve equations that don't have real number solutions, the imaginary unit is denoted by the symbol \(i\). It is uniquely defined by the property \(i^2 = -1\).
This property implies that \(i\) is a square root of \(-1\).
This is different from real numbers, where squares yield non-negative results. The imaginary unit makes it possible to expand the number system into complex numbers, which include combinations of real and imaginary parts, typically expressed as \(a + bi\) where \(a\) and \(b\) are real numbers.

Powers of i

Understanding the powers of \(i\) is crucial for working with complex numbers. The powers of \(i\) cycle every four iterations, as follows:

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

This cycle repeats indefinitely, which means for any integer \(n\), \(i^n\) will equal one of the four values based on the remainder of \(n\) divided by 4.
For example, in the exercise given, \(i^{12}\) simplifies to \(i^0 = 1\) since 12 divided by 4 leaves a remainder of 0.
Understanding this cyclic nature allows you to quickly determine any power of \(i\) without extensive calculation.

Simplifying Expressions

When simplifying expressions involving powers of \(i\), using the cyclical nature of \(i\)'s powers is extremely helpful.
To simplify an expression like \(\frac{1}{i^{12}}\), you begin by recognizing that \(i^{12}\) corresponds to \(i^0\) due to the remainder when dividing 12 by 4.
This results in \(i^{12} = 1\). You then substitute this back into the expression, resulting in \(\frac{1}{1} = 1\).
The key steps include:

• Identifying the power of \(i\).
• Using the cycle to find the equivalent simpler power of \(i\).
• Substitute back to simplify the entire expression.

Mastering these steps ensures you can simplify complex numbers effectively.