Problem 9 - Algebra 2

Question

Simplify. $$ i^{29} $$

Step-by-Step Solution

Answer

The simplified form of \( i^{29} \) is \( i \).

Step 1: Understand the Properties of 'i'

The imaginary unit 'i' is defined as the square root of -1. Thus, we have the following powers of 'i': \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). After \( i^4 \), the powers of 'i' repeat this cycle every four powers.

Step 2: Determine the Cycle of Powers

Recognize that because the powers of 'i' cycle every four, to determine \( i^{29} \), we only need to find the remainder of 29 divided by 4.

Step 3: Divide the Exponent by 4

Calculate 29 divided by 4. The quotient is 7, and the remainder is 1. Hence, \( i^{29} = i^1 \).

Step 4: Apply the Cycle to Simplify

Since \( i^1 = i \), it follows that \( i^{29} = i \).

Key Concepts

Imaginary Unit

The imaginary unit, commonly represented by the symbol 'i', is a fundamental concept in complex numbers. It is defined mathematically as the square root of -1. Since there is no real number that can be squared to equal a negative number, 'i' serves as a placeholder in complex numbers, allowing calculations and solutions for equations where negative roots would otherwise be impossible.\( ewline \)
The rules governing 'i' help simplify expressions involving square roots of negative numbers, and it's vital in fields like engineering, physics, and computer science. When 'i' is used in expressions, it forms what are known as complex numbers, typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(b\)\(i\) represents the imaginary part. This expands our number system beyond real numbers, enabling us to solve a broader range of mathematical problems.

Powers of i

Understanding the powers of the imaginary unit is crucial. As 'i' is defined as \( \sqrt{-1} \), its powers reveal interesting patterns. These powers cycle through specific values, making it predictable.\( ewline \)
Here is a breakdown of several of the initial powers of 'i':

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

After \( i^4 \), the powers repeat this pattern. Thus, \( i^5 \) returns to \( i \), \( i^6 = -1 \), and so on. By knowing just these first four powers, you can ascertain the value of any power of 'i'. It's like having a cheat sheet for any calculations involving 'i'. This cyclic pattern greatly simplifies what could otherwise be a complicated matter.

Cyclic Pattern

The cyclic pattern in the powers of 'i' provides a handy shortcut for calculations. As seen, every fourth power of 'i' brings the cycle back to its beginning: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \). This cycle repeats indefinitely with higher powers, making predictions straightforward.\( ewline \)
For students, this means fewer calculations. Rather than performing extensive multiplication, knowing the cyclic pattern allows you to find any power of 'i' simply by identifying what part of the cycle it lands in. This predictable repetition helps bring order to complex number computations, reducing opportunities for errors and saving time. Remember the pattern and apply it whenever working with powers of 'i'.

Exponent Division

To effectively simplify expressions involving powers of 'i', exponent division becomes an invaluable tool. Since the powers of 'i' cycle every four terms, dividing the exponent by 4 helps determine where in the cycle the power lies.\( ewline \)
Take \( i^{29} \) for example. By dividing 29 by 4, you obtain a quotient of 7 with a remainder of 1. The remainder indicates that \( i^{29} \) aligns with \( i^1 \) in the cycle, resulting in \( i^{29} = i \).\( ewline \)
This method not only simplifies complex equations but also strengthens numerical prediction skills. The technique of dividing the exponent aids in quickly pinpointing the relevant power, helping mathematicians handle complex expressions more efficiently and accurately.