Problem 76 - Algebra A Combined Function

Question

Find each power of \(i\). $$ i^{40} $$

Step-by-Step Solution

Answer

\( i^{40} = 1 \)

Step 1: Understand the Properties of i

The imaginary unit \( i \) has a cyclical pattern in its powers: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). This pattern repeats every four powers. So, every fourth power of \( i \) is a repeat of \( i^4 \) which is 1.

Step 2: Calculate the Remainder of Exponent

To find \( i^{40} \), observe that the powers of \( i \) cycle every 4 turns. Thus, calculate \( 40 \mod 4 \) (the remainder when 40 is divided by 4). The remainder is 0.

Step 3: Use Cyclical Pattern

Since the remainder is 0, we use the fact that \( i^0 = 1 \) to conclude that \( i^{40} = i^{4n} \) (where \( n \) is an integer), specifically \( i^4 = 1 \). Thus, \( i^{40} = (i^4)^{10} = 1^{10} = 1 \).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \( i \), is a basic concept in complex numbers. It's defined by the equation \( i^2 = -1 \). This means that \( i \) is the square root of \(-1\), a number which doesn't exist on the real number line. Instead, it forms the heart of the complex number system. Complex numbers have the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

• \( i^1 = i \) is the base power of \( i \).
• \( i^2 = -1 \) shows its unique property.

These unique properties allow \( i \) to expand our number system to include solutions to equations that do not have solutions within the real numbers, notably those that require the square root of a negative number.

Powers of i

Powers of the imaginary unit \( i \) follow a specific pattern that repeats every four powers. Understanding these powers is crucial when dealing with complex numbers. The first few powers are:

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

After \( i^4 \), the powers simply repeat this cycle. So, \( i^5 \) would revert back to \( i \), \( i^6 \) to \(-1\), and so forth. This repeating pattern is essential in simplifying expressions containing \( i \), ensuring calculations are straightforward and manageable.

Cyclical Pattern in Powers

The cyclical pattern of powers of \( i \) makes calculations much simpler. Because \( i^4 = 1 \), every fourth power equals 1, making it easy to determine higher powers of \( i \). For any integer \( n \), \( i^{4n} = 1 \), a powerful tool when simplifying expressions. For example, to find \( i^{40} \), divide 40 by the cycle length, 4. The operation \( 40 \mod 4 \) results in a remainder of 0. This means \( i^{40} = i^0 \), which equals 1, because \( i^{4n} \) (where \( n \) is an integer) always simplifies to 1. This cyclical behavior is fundamental for working efficiently with powers of \( i \).

Modulus Operation

The modulus operation, denoted as "mod," helps simplify calculations involving powers by identifying the remainder of division. It's crucial when dealing with cyclic patterns, such as powers of \( i \). To apply it, perform division and focus on the remainder:

• Given \( i^{40} \), compute \( 40 \mod 4 \).
• The result is 0, indicating \( i^{40} = i^0 \).
• Since \( i^0 = 1 \) (as \( i^{4n} = 1 \)), \( i^{40} = 1 \).

This operation is a robust method to simplify expressions with cyclic patterns, ensuring you can tackle problems efficiently without getting bogged down by large exponents.