Problem 91 - Elementary and Intermediate Algebra

Question

Simplify each expression. See Example 11 . $$ i^{27} $$

Step-by-Step Solution

Answer

The simplified form of \( i^{27} \) is \( -i \).

Step 1: Understanding the Properties of 'i'

The imaginary unit, often denoted as 'i', is defined by the property that \[ i^2 = -1 \].This property creates a cyclic pattern when powers of 'i' are considered. Specifically, the powers of 'i' repeat every four terms:\[ i^1 = i, \; i^2 = -1, \; i^3 = -i, \; i^4 = 1 \]. After this cycle, the pattern repeats.

Step 2: Finding the Remainder of 27 Divided by 4

To determine which term in the cycle corresponds to \(i^{27}\), we need to find the remainder when 27 is divided by 4 (since the powers of 'i' repeat every 4 terms). Perform the division: \[ 27 \div 4 = 6 \text{ remainder } 3 \]. This means 27 can be expressed as: \[ 27 = 4 \times 6 + 3 \].

Key Concepts

Powers of i

Imaginary numbers often start with understanding the unit imaginary number 'i'. One fundamental property of 'i' is understanding that

• \( i^2 = -1 \)

This property defines 'i' as the square root of -1, which is the foundation of complex numbers. Now, when we consider powers of 'i', an interesting pattern emerges. Each incremental power of 'i' produces a different outcome:

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

Once we reach \( i^4 \), the powers of 'i' start to repeat themselves, forming a reliable cycle. Grasping these powers is essential for simplifying expressions in imaginary numbers, which often appears in algebraic problems and equations. It's critical to recognize these cyclical patterns as they form the backbone of imaginary number manipulation.

Cyclic Pattern

The powers of 'i' demonstrate a cyclic pattern that repeats every four terms. This is a cycle that every student should memorize as it simplifies complex calculations involving 'i'. Let's break it down:

• The sequence begins with: \( i, -1, -i, 1 \)
• Then, it continues again: \( i^{5}= i, i^{6}= -1 \), and so on.
• Recognizing this four-term cycle is crucial in simplifying expressions efficiently.

For example, if ever faced with a problem like \( i^{27} \), knowing the cyclic pattern saves time. Identifying where \( i^{27} \) falls in the cycle involves using calculations from the pattern. Since the sequence resets every four numbers, determining cycle placement helps simplify otherwise bulky algebraic expressions. This cyclical nature is thus a key part of mastering imaginary numbers.

Modular Arithmetic

Modular arithmetic plays a vital role when working with powers of 'i'. This framework helps find where a power falls within the established cycle. To determine the equivalent power of a large exponent like \( i^{27} \), modular arithmetic comes into play.
The initial step is performing a division:

• Divide 27 by 4, the cycle length.
• We calculate: \( 27 \div 4 = 6 \) with a remainder of 3.

This remainder is crucial, as it indicates the position within the cycle — in this instance, the third position. Therefore, \( i^{27} \) is equivalent to \( i^3 \). Since we know \( i^3 = -i \), it's easy to simplify: \( i^{27} = -i \).

This method reliably reduces any power of 'i' into a manageable form by finding its equivalent remainder. Understanding and mastering modular arithmetic helps streamline calculations, making complex number problems less intimidating.