Problem 91 - Intermediate Algebra

Question

Simplify each expression. $$ i^{27} $$

Step-by-Step Solution

Answer

The simplified expression is \(-i\).

Step 1: Understanding Powers of i

Remember that the imaginary unit \( i \) has a cyclical pattern when raised to powers. The main powers to know are: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). These repeat every four powers, so \( i^5 = i \), \( i^6 = -1 \), and so on.

Step 2: Determine Cycle Completion

Since the pattern repeats every 4 powers, find the remainder when 27 is divided by 4. This helps identify the cycle position of \( i^{27} \). Calculate: \( 27 \div 4 = 6 \) remainder \( 3 \), meaning \( i^{27} = i^3 \).

Step 3: Simplify Using the Cycle

We determined that \( i^{27} = i^3 \). From our knowledge of \( i \)'s powers:\( i^3 = -i \). So, \( i^{27} = -i \).

Key Concepts

Imaginary Unit

The imaginary unit, usually denoted as \( i \), is a fundamental concept in complex numbers. It is defined as the square root of -1: \( i = \sqrt{-1} \). This definition is not applicable to real numbers since no real number squared gives a negative result. However, in the realm of complex numbers, \( i \) unlocks the possibility for solving equations where the square of a number is negative.

The introduction of \( i \) allows us to extend traditional algebra into complex numbers. This is crucial for various applications in engineering, physics, and advanced mathematics. By treating \( i \) as the unit of the imaginary part of a complex number, we can express complex numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Powers of i

The powers of the imaginary unit \( i \) follow a cyclical pattern that continuously repeats every four exponents. Understanding this cycle is key to simplifying expressions involving powers of \( i \). Here's the sequence:

• \( i^1 = i \): This is simply the imaginary unit itself.
• \( i^2 = -1 \): Multiplying \( i \) by itself results in \(-1\).
• \( i^3 = -i \): This is found by multiplying \( i^2 \) by \( i \) again, resulting in \(-i\).
• \( i^4 = 1 \): Multiplying \( i^3 \) by \( i \) gives 1, completing the cycle.

After every four exponents, the pattern restarts, meaning \( i^5 = i \), \( i^6 = -1 \), and so forth. By memorizing this cycle, you can quickly determine the outcome of any power of \( i \). This cyclic behavior simplifies the process of working with complex numbers and allows for easier calculations in both homework and practical applications.

Cyclical Pattern of Powers

The cyclical pattern of powers for the imaginary unit \( i \) is a handy mathematical property that reduces the complexity of calculations. To employ this pattern, apply these simple steps:

• Identify the exponent of \( i \) you are dealing with. For instance, in \( i^{27} \).
• Determine the cycle position by dividing the exponent by 4 and finding the remainder. This step helps pinpoint which point of the cycle is equivalent. For example, \( 27 \div 4 = 6 \) remainder \( 3 \), which means \( i^{27} = i^3 \).
• Refer back to the established cycle to find the value. Since \( i^3 = -i \), it follows that \( i^{27} = -i \).

This method of expressing powers using the cyclical nature effectively simplifies what might otherwise be an intimidating calculation. By practicing this cycle recognition, you can master simplifying complex numbers with ease.