Problem 56 - Algebra 2

Question

Simplify. $$ i^{38} $$

Step-by-Step Solution

Answer

The simplified form of \( i^{38} \) is \(-1\).

Step 1: Understand the properties of 'i'

The imaginary unit 'i' has the property that \( i^2 = -1 \). Hence, powers of 'i' cycle in a sequence every four: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), \( i^5 = i \), and so on.

Step 2: Determine the power modulo 4

To simplify \( i^{38} \), calculate \( 38 \mod 4 \). When 38 is divided by 4, the remainder is 2, as 38 divided by 4 is 9 with a remainder of 2.

Step 3: Apply the cycle property

Since the remainder when 38 is divided by 4 is 2, \( i^{38} \) is equivalent to \( i^2 \) in the cycle of powers of 'i'. By the properties of 'i', \( i^2 = -1 \).

Key Concepts

Imaginary Unit

The imaginary unit, commonly denoted as "i," is a cornerstone concept in the field of complex numbers. Unlike real numbers, which lie on the number line, imaginary numbers are expressed in terms of the square root of negative one. The defining property of the imaginary unit is that \( i^2 = -1 \). This unique property allows mathematicians and engineers to perform calculations that model phenomena in two-dimensional spaces, among many other applications. Imaginary numbers extend the concept of numbers beyond the real line, making it possible to solve equations that would otherwise be unsolvable. For instance, the equation \( x^2 + 1 = 0 \) does not have a solution in real numbers but can be solved using the imaginary unit where \( x = i \). Imaginary numbers are not just figments of mathematical creativity; they are very real and useful in complex number systems that represent concepts like electrical engineering currents and wave functions in quantum mechanics.

Powers of i

Once you understand the imaginary unit and its foundational role, knowing the powers of \( i \) becomes the next step. The powers of \( i \) follow a predictable cycle every four exponents:

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

After \( i^4 \), the cycle repeats. So, learning this sequence of the first four powers enables you to quickly evaluate any power of \( i \). This is crucial for simplifying expressions that involve powers of \( i \), like \( i^{38} \).By recalling this four-step cycle, you can easily deduce that \( i^{38} \), calculated with morning math, simplifies to \( i^2 \), which equals \(-1\). Tracking this relationship helps solve many problems in complex number theory, providing efficient computational steps for larger power reductions.

Modulo Operation

The modulo operation is a helpful mathematical concept that simplifies calculations, particularly in the context of powers of \( i \). The operation involves dividing one number by another and finding the remainder. For example, to simplify \( i^{38} \), you calculate \( 38 \mod 4 \).Using division, you determine how many times 4 fits into 38 and what remains. In this case, 4 fits into 38 a total of 9 times, which uses up 36 of the number, leaving a remainder of 2. Therefore, \( 38 \mod 4 = 2 \).This remainder is crucial because it corresponds directly to the power of \( i \) needed to simplify the expression. Since the powers of \( i \) follow a cycle of four, \( i^{38} \) simplifies to \( i^2 \), both being the remainder and the final value. This operation is widely used, not just in complex numbers, but in various fields, such as computer science, where cyclic behavior is often modeled.