Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Write the quotient in standard form. $$\frac{1}{(2 i)^{3}}$$

Step-by-Step Solution

Answer

\(\frac{1}{8}i\)

Step 1: Simplify the Complex Number

Since \(i^{2} = -1\), the given complex number (2i) to the power of 3 can be simplified as \((2 i)^{3} = 2^{3} \cdot i^{3}=8 i^{3}=-8i\). The reason for this is that \(i^{3}=i^{2}*i=(-1)*i=-i\).

Step 2: Find Reciprocal of the Simplified Complex Number

To find the reciprocal of a complex number, we have to divide 1 by the complex number. So the reciprocal of -8i is \(\frac{1}{-8i}=-\frac{1}{8i}\).

Step 3: Write the Number in Standard Form

Multiply both numerator and denominator by \(i\) to get rid of \(i\) in the denominator. This results in \( -\frac{1 \cdot i}{8i \cdot i} =-\frac{1 \cdot i}{-8} = \frac{1}{8}i \).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit, denoted as \(i\), is foundational. It is defined such that \(i^2 = -1\). This unique trait means that \(i\) represents the square root of negative one. Imaginary numbers are essential in solving equations that have no real solutions, such as those involving the square root of negative numbers. For example, \( \sqrt{-4} = 2i \).

• The imaginary unit helps extend our number system to include complex numbers, which are numbers in the form \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
• Mathematically, \(i^3 = i \times i^2 = i \times -1 = -i\), and \(i^4 = i \times i \times i \times i = 1\).

Understanding \(i\) and its powers is crucial when working with complex numbers, especially when raising \(i\) to higher powers or simplifying expressions involving \(i\). It often involves recognizing repeating patterns, as every four powers, the sequence \(i, -1, -i, 1\) repeats, simplifying calculations significantly.

Reciprocal

The concept of a reciprocal involves flipping a number to "undo" multiplication, essentially interchange the numerator and denominator. It is applicable to all non-zero numbers, including complex numbers. For a real number \(a\), its reciprocal is \(\frac{1}{a}\). When dealing with complex numbers, the concept is similar.

• If \(z = x + yi\), its reciprocal is given as \(\frac{1}{z}\).
• To simplify, multiply numerator and denominator by the complex conjugate (changing the sign of the imaginary part) to eliminate the imaginary unit in the denominator.

In the provided exercise, the reciprocal calculation simplifies \(-8i\) to \(-\frac{1}{8i}\), which is complex since it keeps \(i\) in the denominator. To further simplify, multiply by \(i\) to move the imaginary unit to the numerator, making the expression easier to handle and eventually bringing it to standard form.

Standard Form

Complex numbers are generally expressed in standard form, which is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This form is the most common way to present complex numbers because it neatly separates the real and imaginary components.

• The standard form aims to simplify the expression and provide a unified representation to make arithmetic operations easier.
• When converting a complex expression, like \(\frac{1}{-8i}\), to standard form, it's key to remove any imaginary parts from the denominator as seen in the exercise.

To achieve this, we multiply by \(i\) to have \(\frac{i}{-8}\), leading ultimately to \(\frac{1}{8}i\). This result fits perfectly into the standard form where the real part (\(a\)) is zero, leading to an expression of form \(0 + \frac{1}{8}i\), emphasizing that it's purely imaginary.