Problem 47 - College Algebra

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 \(3i\).

Step 1: Simplify the Expression

Begin by recognizing that the expression involves powers of the imaginary unit, \( i \). Recall that \( i^2 = -1 \) and subsequently \( i^3 = i^2 \cdot i = -i \) and \( i^4 = 1 \). The original expression is \( \frac{1}{i} + \frac{4}{i^3} \).

Step 2: Rationalize the Denominator for \( \frac{1}{i} \)

Rewrite \( \frac{1}{i} \) by multiplying the numerator and denominator by \( i \): \[ \frac{1 \cdot i}{i \cdot i} = \frac{i}{-1} = -i. \]

Step 3: Rationalize the Denominator for \( \frac{4}{i^3} \)

Since \( i^3 = -i \), rewrite \( \frac{4}{i^3} \) as \( \frac{4}{-i} \). Multiply the numerator and the denominator by \( i \) to get: \[ \frac{4 \cdot i}{-i \cdot i} = \frac{4i}{-(-1)} = 4i. \]

Step 4: Combine the Complex Expression

Now combine the results from Steps 2 and 3: \(-i + 4i\). Simplify the expression by combining the terms: \(-i + 4i = 3i.\)

Step 5: Final Verification

Verify the simplifications step-by-step to ensure accuracy: The powers of \( i \) were correctly simplified and the combined expression correctly evaluated.

Key Concepts

Imaginary Unit

Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part involves the imaginary unit, denoted as \( i \). The imaginary unit \( i \) is defined by the property: \[ i^2 = -1. \] This means when you square \( i \), you get \(-1\).
When dealing with imaginary numbers, remember the following powers of \( i \):

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

These powers repeat every four steps, so \( i^5 = i \), and so on. This cycle is crucial for simplifying terms in expressions involving high powers of \( i \).
Understanding \( i \) and its properties helps in manipulating complex number expressions effectively. Always simplify the powers of \( i \) when needed to make calculations clear and manageable.

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate imaginary numbers from the denominator of a fraction. This process makes expressions easier to read and work with. The denominator is made 'rational,' meaning that it no longer contains the imaginary unit \( i \).
Consider a fraction with \( i \) as the denominator, such as \( \frac{1}{i} \). To rationalize, multiply both numerator and denominator by \( i \): \[ \frac{1 \cdot i}{i \cdot i} = \frac{i}{-1} = -i. \] This operation uses the fact that \( i^2 = -1 \), turning the denominator into an integer, thus eliminating \( i \).
This technique is beneficial for any fraction where the denominator is an imaginary unit or includes a power of \( i \). Rationalizing allows you to transform the expression into a form that's easier to add, subtract, multiply, or divide with other numbers.

Powers of i

When dealing with expressions involving complex numbers, understanding the behavior of different powers of the imaginary unit \( i \) is essential. Each power of \( i \) follows a specific pattern:

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

These four values cycle repeatedly, helping to simplify expressions with powers of \( i \). For example, in the expression \( \frac{4}{i^3} \), recognizing that \( i^3 = -i \) allows you to rewrite and simplify it.
Knowing this cycle enables you to immediately interpret and reduce problems involving powers higher than 4, like \( i^5 = i \), \( i^6 = -1 \), and so forth. This makes computations more straightforward and less error-prone.
Always remember and apply this pattern when simplifying complex expressions in mathematics.