Problem 48 - College Algebra

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{1}{i^{11}}-\frac{1}{i^{21}} $$

Step-by-Step Solution

Answer

2i

Step 1: Simplify Power of i

First, recognize the pattern of powers of the imaginary unit, \( i \): - \( i^1 = i \)- \( i^2 = -1 \)- \( i^3 = -i \)- \( i^4 = 1 \)This pattern repeats every four powers. Calculate the powers for the terms \( i^{11} \) and \( i^{21} \): Since 11 mod 4 is 3, \( i^{11} = i^3 = -i \). Since 21 mod 4 is 1, \( i^{21} = i^1 = i \).

Step 2: Rewrite the Expression

Substitute \( i^{11} = -i \) and \( i^{21} = i \) back into the original expression:\[ \frac{1}{i^{11}} - \frac{1}{i^{21}} = \frac{1}{-i} - \frac{1}{i} \].

Step 3: Simplify the Fractions

Multiply the numerator and the denominator of each fraction by \( i \) to simplify and get rid of the imaginary denominator:For \( \frac{1}{-i}\):\[ \frac{1}{-i} \cdot \frac{i}{i} = \frac{i}{-i^2} = \frac{i}{1} = i \].For \( \frac{1}{i}\):\[ \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \].

Step 4: Subtract the Simplified Fractions

Simplify the original expression with the newly calculated terms:\[ i - (-i) = i + i = 2i \].

Step 5: Final Simplified Complex Number

The expression simplifies to:\[ 2i \].This is the simplest form of the complex number representing the initial expression.

Key Concepts

Powers of Imaginary Unit

When dealing with complex numbers, understanding the powers of the imaginary unit (denoted as \( i \)) is crucial. The imaginary unit \( i \) satisfies the equation \( i^2 = -1 \). From this relationship, a repeating pattern can be observed in the powers of \( i \). This pattern is:

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

This cycle repeats every four powers. To determine the value of \( i \) raised to any power, you simply calculate the remainder when the exponent is divided by 4. For example, \( i^{11} \) can be computed by dividing 11 by 4, which gives a remainder of 3; hence, \( i^{11} = i^3 = -i \). Similarly, \( i^{21} \) results in a remainder of 1, meaning \( i^{21} = i^1 = i \). Knowing this pattern allows for quick evaluations of complex numbers raised to large exponents.

Simplifying Complex Expressions

Simplifying complex expressions often involves dealing with powers of \( i \) and managing complex fractions. When faced with expressions containing the imaginary unit, especially in the denominator, a common approach is to multiply the numerator and the denominator by the conjugate or by the imaginary unit itself. This technique helps eliminate imaginary numbers from the denominator.

In our case, the expression \( \frac{1}{i^{11}} - \frac{1}{i^{21}} \) simplifies to \( \frac{1}{-i} - \frac{1}{i} \). Multiplying both fractions by \( i \), we simplify them as follows:

• For \( \frac{1}{-i} \), multiply numerator and denominator by \( i \): \( \frac{1}{-i} \cdot \frac{i}{i} = \frac{i}{-i^2} = \frac{i}{1} = i \).
• For \( \frac{1}{i} \), perform the same operation: \( \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \).

With these simplifications, the complex expression becomes \( i - (-i) = i + i = 2i \), resulting in a neat and simplified expression.

Imaginary Unit Properties

The imaginary unit \( i \) is a fundamental building block in complex number arithmetic. One of its most essential properties is that \( i^2 = -1 \). This counterintuitive property opens the door to complex numbers, as it allows for operations that extend beyond real numbers.

Some important properties to remember about \( i \):

• Multiplying \( i \) by itself repeatedly yields alternating results: \( i, -1, -i, 1 \). This completes the cycle.
• The concept of \( i \) originates from the need to solve equations like \( x^2 + 1 = 0 \), where real number solutions don't exist.
• When handling complex numbers, always aim to simplify using \( i^2 = -1 \) to convert expressions to a more recognizable form.

Understand and use these imaginary unit properties to simplify and manipulate complex expressions effectively. This familiarity is key to mastering topics within complex number theory and beyond.