Problem 73 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{5}$$

Step-by-Step Solution

Answer

\(i^5 = i\).

Step 1: Understand the Cycle of Powers of i

The imaginary unit, represented as \(i\), is defined such that \(i^2 = -1\). The powers of \(i\) follow a repeating cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats for higher powers.

Step 2: Determine the Remainder of the Division by 4

Since the powers of \(i\) repeat every 4, we can use the remainder when dividing the exponent by 4 to find the equivalent lower power of \(i\). For \(i^5\), divide 5 by 4, which gives us a quotient of 1 and a remainder of 1.

Step 3: Use the Remainder to Simplify the Power of i

The remainder of 1 from Step 2 implies that \(i^5\) is equivalent to \(i^1\) based on the cycle. Therefore, \(i^5 = i^1\).

Key Concepts

Powers of i

The powers of the imaginary unit, denoted as \(i\), exhibit a fascinating cyclical pattern. This cycle is crucial in simplifying expressions involving powers of \(i\). Knowing the fundamental definition of \(i\) is a good starting point: \(i^2 = -1\). From this foundation, a predictable pattern unfolds:

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

After \(i^4 = 1\), the cycle repeats itself, meaning that \(i^5\) will be the same as \(i^1\), \(i^6\) the same as \(i^2\), and so forth. A neat trick to simplify any power of \(i\) is to divide the power by 4, and use the remainder to find the corresponding power within the cycle. This cycle repetition helps in quickly solving problems without having to multiply \(i\) endlessly.

Complex Numbers

Complex numbers extend the idea of the standard number line. They are essentially a combination of real numbers and imaginary numbers. In a complex number, the structure is typically shown as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) representing the square root of -1.
Complex numbers enable us to solve equations that have no real solutions, providing a more complete numerical system.The concept of complex numbers is vital in various fields, including physics and engineering, as it allows for the representation of concepts like waveforms and electronic currents. When visualizing complex numbers, they are often plotted on the complex plane, where the horizontal axis represents real values and the vertical axis represents imaginary values. This dual representation can help in visualizing complex operations like addition and multiplication.

Algebra

Algebra is one of the fundamental building blocks of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express real-world questions in a form that can be solved to find unknown values. When we expand algebra to include imaginary numbers and thus complex numbers, it enables a wider range of applications and solutions.Understanding the algebraic handling of imaginary numbers, like simplifying powers of \(i\), is a crucial skill. This includes:

• Recognizing the cyclical nature of \(i\)'s powers, as discussed in the powers of \(i\) section
• Learning to manipulate and simplify expressions involving \(i\)
• Applying division and modular arithmetic, like dividing the exponent by 4, to determine the equivalent power of \(i\)

By mastering these techniques, one can tackle a wide array of mathematical problems that involve complex and imaginary numbers, enhancing both problem-solving skills and numerical literacy.