Problem 81 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{9}}$$

Step-by-Step Solution

Answer

\( \frac{1}{i^9} = -i \)

Step 1: Understand Powers of i

The powers of the imaginary unit \( i \) repeat every four cycles: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). These are the base cycles for powers of \( i \). Notice that powers of \( i \) beyond \( i^4 \) repeat according to this cycle.

Step 2: Simplify the Exponent

We need to express \( i^9 \) in terms of the base cycles. Since the powers of \( i \) repeat every four cycles, we find the remainder of the exponent when divided by 4: \( 9 \div 4 = 2 \) remainder 1. This means \( i^9 = i^{4\times2 + 1} = (i^4)^2 \cdot i^1 = 1^2 \cdot i = i \).

Step 3: Simplify the Expression

Now that we know \( i^9 = i \), substitute \( i \) back into the expression: \( \frac{1}{i^9} = \frac{1}{i} \). To simplify \( \frac{1}{i} \), multiply numerator and denominator by \( i \), yielding \( \frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \).

Key Concepts

powers of i

Complex numbers often involve the use of the imaginary unit, denoted as \( i \). Understanding the powers of \( i \) is crucial for simplifying expressions that involve complex numbers. The key to mastering the powers of \( i \) is recognizing their cyclical nature. The powers of \( i \) repeat every four steps:

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

After these four powers, the sequence starts again. For instance, \( i^5 = i^1 \), \( i^6 = i^2 \), and so on. This repeating cycle allows you to simplify any power of \( i \) by dividing the exponent by 4 and using the remainder to identify the equivalent power from the cyclical sequence. This approach helps us manage expressions like \( i^9 \), where dividing 9 by 4 gives a remainder of 1, matching it with \( i^1 = i \).

imaginary unit

The imaginary unit \( i \) is a fundamental component in the study of complex numbers. It is defined by the property that \( i^2 = -1 \). This elusive quality allows complex numbers to extend beyond the real number line into a new dimension, providing solutions to equations that have no real solutions, like \( x^2 = -1 \).
Imaginary numbers paired with real numbers form complex numbers, expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers. The real part, \( a \), and the imaginary part, \( bi \), together comprise the complex number. This combination enables a comprehensive approach to problems involving imaginary units, as they simplify the orchestration of expressions and provide additional ways to interpret numerical data.

simplifying expressions

Simplifying expressions involving complex numbers, especially those with the imaginary unit \( i \), often requires several steps to ensure clarity and correctness. Expressions like \( \frac{1}{i} \) can be initially tricky but become manageable with some algebraic manipulation.
To simplify \( \frac{1}{i} \), multiply both the numerator and denominator by \( i \). This maneuver exploits the property of \( i^2 = -1 \), leading to:

• Numerator: \( 1 \times i = i \)
• Denominator: \( i \times i = i^2 = -1 \)

This transforms the expression into \( \frac{i}{-1} \), which simplifies further to \(-i \). Such techniques not only clarify the expression but also reveal deeper insights into the nature of complex numbers, demonstrating how the challenges posed by \( i \) can be resolved with basic algebraic strategies.