Problem 90 - Intermediate Algebra

Question

Simplify each expression. $$ i^{19} $$

Step-by-Step Solution

Answer

\(i^{19} = -i\).

Step 1: Identify the Pattern of Powers of i

The imaginary unit, denoted as \(i\), has a repeating cycle of powers. \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats every four powers.

Step 2: Determine the Remainder

Since the powers of \(i\) repeat every 4, divide the exponent by 4 to find the remainder. For \(i^{19}\), compute \(19 \div 4\) which gives a quotient of 4 and a remainder of 3.

Step 3: Simplify Using the Remainder

The remainder tells you which power of the cycle corresponds to \(i^{19}\). Since the remainder is 3, \(i^{19}\) is equivalent to \(i^3\). From the pattern identified earlier, \(i^3 = -i\).

Key Concepts

Powers of i

When dealing with complex numbers, understanding the powers of the imaginary unit, denoted as \(i\), is fundamental. The imaginary unit \(i\) is defined by its property that \(i^2 = -1\). This property is the cornerstone of powers of \(i\) and leads to a repeating cycle:

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

After \(i^4\), the cycle starts over: \(i^5 = i\), \(i^6 = -1\), and so on. Notice the cycle length is 4, which means after every 4th power, the results repeat in the same order.
Hence, to calculate any power of \(i\), you can use this cycle. Simply divide the exponent by 4 and use the remainder for rapid calculation. For instance, with \(i^{19}\), compute \(19 \div 4\) to find a remainder of 3, which tells us \(i^{19} = i^3 = -i\). This simplifies computations for any power of \(i\).

Imaginary Unit

The imaginary unit, represented by \(i\), is an essential component of complex numbers. Although the name 'imaginary' might imply something isn't real, \(i\) is very much a part of mathematics. The defining property of \(i\) is \(i^2 = -1\).
This property is what separates \(i\) from real numbers, as no real number has this property. It allows us to extend the concept of numbers to complex numbers, which involve both real and imaginary parts.
Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, and \(b\) is the imaginary part. The presence of \(i\) in complex numbers enables the solution of equations that wouldn't have solutions in the realm of real numbers alone, such as \(x^2 + 1 = 0\).
Complex numbers and \(i\) have vast applications across areas including engineering, physics, and signal processing, showing how this 'imaginary' concept forms the foundation for solving real-world problems.

Simplifying Expressions

Simplifying expressions with complex numbers, especially those involving powers of \(i\), can initially appear complex. However, recognizing patterns and understanding the properties of \(i\) can significantly ease this task.
Use the repeating cycle of \(i\) (every 4th power) to your advantage. Simplification generally involves the following steps:

• Identify the exponent you've been given.
• Divide the exponent by 4 to discern the remainder.
• Refer to the cycle of \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\) to find the equivalent simpler form.

For example, if asked to simplify \(i^{19}\) – find the remainder by dividing 19 by 4, which gives 3. Therefore, \(i^{19} = i^3 = -i\).
This method is highly effective not only for simple expressions like \(i^{19}\), but for any powers of \(i\), assuring you handle them correctly and swiftly. By routinely practicing these steps, you'll find that working with complex numbers becomes more intuitive.