Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{1}{i}+\frac{4}{i^{3}} $$

Step-by-Step Solution

Answer

The simplified complex number is \(-5i\).

Step 1: Rewrite Using Powers of i

Recall that higher powers of the imaginary unit can be simplified using the identities: \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Therefore, \(i^3 = -i\). Use these identities to rewrite the expression:\[\frac{1}{i} + \frac{4}{i^3} = \frac{1}{i} + \frac{4}{-i}\]

Step 2: Multiply Numerator and Denominator by i

To simplify \(\frac{1}{i}\), multiply both the numerator and the denominator by \(i\):\[\frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i\]

Step 3: Multiply Numerator and Denominator by -i

Similarly, for \(\frac{4}{-i}\), multiply both the numerator and the denominator by \(-i\):\[\frac{4}{-i} \times \frac{-i}{-i} = \frac{-4i}{-i^2} = \frac{-4i}{1} = -4i\]

Step 4: Combine the Terms

Combine the terms obtained from simplification:\[-i - 4i = -5i\]

Step 5: Final Simplified Complex Number

The final expression in the form \(a + bi\) is:\[0 - 5i\]Thus, the simplified complex number is \(-5i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined such that \(i^2 = -1\). This definition allows us to work with square roots of negative numbers, which are not possible in the set of real numbers. The introduction of the imaginary unit extends the real number system to form what is known as the complex number system.
Using the imaginary unit, any negative square roots can be expressed in terms of \(i\). For instance, the square root of \(-4\) is \(2i\), because when you square \(2i\), you get \(4i^2\), and since \(i^2 = -1\), it's \(4(-1) = -4\). This essential building block allows for more comprehensive solutions to equations that don't have real number solutions.

Powers of i

The powers of \(i\) exhibit a cyclical pattern, repeating every four powers. Understanding this repeating cycle is crucial when working with expressions involving \(i\).
Here is the pattern:

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

After that, the cycle repeats: \(i^5 = i\), \(i^6 = -1\), and so on. Recognizing this pattern helps simplify complex expressions and solve equations more easily. For example, if you encounter \(i^3\) in a problem, you can directly replace it with \(-i\), simplifying computations significantly.

Simplifying Expressions

When dealing with complex numbers and expressions that include \(i\), simplifying them helps turn the expression into a more manageable form. One common technique is eliminating \(i\) from the denominator.
Consider the expression \(\frac{1}{i}\). To simplify, multiply both the numerator and the denominator by \(i\), which yields \(-i\), as shown in the solution. Similarly, for \(\frac{4}{i^3}\), substituting \(i^3 = -i\) and then performing multiplication to remove \(i\) from the denominator results in \(-4i\). This process makes arithmetic simpler and leads to combining like terms more straightforwardly.

Complex Number Form

A complex number is typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. This form helps easily recognize and categorize complex numbers.
In the given problem, after simplifying the expressions, we ended up with \(-5i\). This result can be viewed as a complex number with a real part of 0 (since no real numbers are included) and an imaginary part of \(-5i\). Writing complex numbers in this standardized form aids in performing operations, such as addition or multiplication, and assures consistency across mathematical solutions.