Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ i^{7}\left(1+i^{2}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Simplify Powers of i

The imaginary unit \( i \) follows a cycle with powers: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). This cycle repeats. For \( i^7 \), determine its equivalent by considering its position in the cycle. Since \( 7 \mod 4 = 3 \), we have \( i^7 = i^3 = -i \).

Step 2: Simplify the Expression in Parentheses

In the expression \( 1 + i^2 \), substitute \( i^2 = -1 \). Therefore, \( 1 + i^2 = 1 + (-1) = 0 \). This implies the entire expression becomes zero.

Step 3: Multiply the Simplified Components

Since \( i^7 = -i \) and \( 1 + i^2 = 0 \), multiply these results together: \( -i \times 0 = 0 \).

Step 4: Write in Simplified Complex Form

In complex number terms, 0 is already simplified, so the result is \( 0 + 0i \). However, it can simply be expressed as \( 0 \).

Key Concepts

Simplifying Complex Expressions

Complex expressions often involve operations on numbers that include both a real and an imaginary part. For clarity, let's start with understanding what a simplified complex number is. Typically, a complex number is given in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. When simplifying complex expressions, our goal is to write the expression as simply as possible without losing its mathematical integrity.

Here are some key steps:

• Combine like terms.
• Substitute known values (like \( i^2 = -1 \)).
• Perform operations separately on real and imaginary parts.

Breaking down a complex expression, as seen in the exercise \( i^{7}(1+i^{2}) \), involves simplifying powers of the imaginary unit and reviewing operations within the parentheses. Remember that simplifying includes reducing any unnecessary parentheses and ensuring operations adhere to mathematical rules. With practice, simplifying becomes intuitive. Remember, any complex number where \( b = 0 \) is essentially a real number, just as seen in the result \( 0 \).

Powers of the Imaginary Unit

The imaginary unit \( i \) is fundamental to understanding complex numbers. A peculiar but fascinating aspect of \( i \) is its powers, which repeat cyclically every four terms:

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

This cycle repeats indefinitely, meaning \( i^5 = i \), \( i^6 = -1 \), and so forth.

To simplify higher powers, determine the remainder when the exponent is divided by 4, as the exponent shares its position in the cycle. For example, in the exercise \( i^7 \):
1. Divide 7 by 4, giving a remainder of 3.
2. Thus, \( i^7 = i^3 = -i \).

Understanding these cycles helps streamline complex arithmetic, making problems like \( i^7 \) faster to solve.

Complex Number Arithmetic

Arithmetic with complex numbers involves operations such as addition, subtraction, multiplication, and division. These are not much different from operations with real numbers, though they follow specific rules due to the inclusion of \( i \).

A few quick rules:

• Addition/subtraction: Combine like terms. Real with real, imaginary with imaginary.
• Multiplication: Distribute terms, considering \( i^2 = -1 \).
• Division: Involves multiplying numerator and denominator by the conjugate to eliminate \( i \) from the denominator.

In our provided example, after simplifying initial expressions, multiplying gives us \(-i \times 0\), which results in \(0\) as any number times zero is zero.

This illustrates that despite initially complex appearances, results can be straightforward. Approach complex arithmetic steps methodically and ensure all parts, like simplifying \( i^2 \), stay correct. With practice, these arithmetic operations become second nature, enabling you to solve more complex expressions confidently.