Problem 71 - Algebra A Combined Function

Question

Find each power of \(i\). $$ i^{8} $$

Step-by-Step Solution

Answer

\(i^8 = 1\)

Step 1: Basic Multiplication Properties of i

The imaginary unit \(i\) has the property \(i^{2} = -1\). Using this, we can derive other powers of \(i\) by repeated multiplication.

Step 2: Find Basic Powers of i

Let's establish the cycle of powers for \(i\):- \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = i^2 \cdot i = -1 \cdot i = -i\)- \(i^4 = i^3 \cdot i = -i \cdot i = 1\)Notice the cycle: every four powers, \(i\) returns to 1. This is a key observation for finding higher powers.

Step 3: Determine the Cycle for i to the 8th Power

Since the powers of \(i\) cycle every 4 powers, we can calculate higher powers by dividing the exponent by 4 and examining the remainder. For \(i^8\), divide 8 by 4:\[8 \div 4 = 2 \text{ with a remainder of } 0\]When the remainder is 0, the power corresponds to \(i^4 = 1\).

Step 4: Calculate i to the 8th Power

Using the cycle determined in Step 3, since the remainder was 0, we know that \(i^8 = 1\), consistent with the pattern where \(i^4 = 1\).

Key Concepts

powers of i

When studying complex numbers, understanding the powers of \(i\), the imaginary unit, is fundamental. The imaginary unit \(i\) has a defining property: \(i^2 = -1\). Leveraging this property helps us discover higher powers of \(i\) through multiplication. However, memorizing every power can become cumbersome, so recognizing patterns can greatly simplify the process. Let's explore some basic powers of \(i\):

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

Notice how after every fourth power \(i\) returns to \(1\). This repeating cycle is essential for computing any power of \(i\). Understanding this can help solve complex equations and simplify computations.

imaginary unit

The concept of the imaginary unit is pivotal in the realm of complex numbers. Denoted as \(i\), it is defined by the property \(i^2 = -1\). While real numbers are perfectly adequate for many applications, there are situations, especially in engineering and physics, where the introduction of an imaginary component helps solve problems otherwise unsolvable.

The imaginary unit enables mathematicians and engineers to extend the real number line into the complex plane, allowing for complex arithmetic operations involving both real and imaginary parts. This is particularly useful for representing phenomena like alternating currents or waveforms in electrical engineering or quantum mechanics.

• Visualization: Imaginary numbers appear on a perpendicular axis to real numbers in the complex plane.
• Utility: Useful in solving quadratic equations with negative discriminants and performing Fourier transforms.
• Application: Prominent in advanced signal processing and control systems.

cyclic nature of i

The cyclic nature of \(i\) is a fascinating subject. It brings about a predictable pattern in its powers, making it simpler to work with complex number operations. As previously discussed, \(i\)'s powers exhibit a cycle every four terms: \(i, -1, -i, 1\). This cyclic behavior makes calculating high powers straightforward.

To find powers such as \(i^8\) or \(i^{24}\), you need to perform simple division by 4 and consider the remainder:

• If remainder is 0, the result is 1 (as in \(i^4\)).
• If remainder is 1, the result is \(i\).
• If remainder is 2, the result is \(-1\).
• If remainder is 3, the result is \(-i\).

This cyclic nature, thus, provides an elegant way to simplify calculations involving powers of \(i\), ensuring consistency and predictability when working with complex numbers. Understanding this concept is powerful in fields such as electronics and advanced mathematics.