Problem 52 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The quotient in standard form is \(-i\).

Step 1: Cube the denominator

In order to find the cube of a complex number, apply the given exponent to both real and imaginary parts of the complex number, so \((3i)^3\ = 3^3 \cdot i^3\ = 27i^3\), but since \(i^3\ = -i\) (this is a property of imaginary numbers), then \((3i)^3\ = -27i\).

Step 2: Reciprocal of the denominator

Find the reciprocal of the complex number in the denominator, which is \(-27i\). The reciprocal of a number is just 1 divided by the number. However when dealing with complex numbers, we use the concept of a conjugate. We multiply and divide by the conjugate of \(-27i\) which is \(27i\), so the reciprocal will be \(\frac{1}{-27i} \cdot \frac{27i}{27i}=-i\). If the denominator was a complex number \((a+bi)\) the conjugate would be \(a-bi\).

Step 3: Multiplication

Finally, compute the product of 1 and the reciprocal of the cube of \(3i\), which is \(-i\). So the quotient is \(1 \cdot -i = -i\).

Key Concepts

Imaginary Unit

Complex numbers introduce a fascinating concept called the "Imaginary Unit." Denoted as \(i\), this unit is defined by the property that \(i^2 = -1\). Imaginary numbers stem from the necessity of having solutions to equations that do not have real number answers. For example, the square root of a negative number, \(\sqrt{-1}\), is expressed as \(i\). This forms the backbone of complex numbers, marrying real numbers with imaginary numbers to offer a more comprehensive number system known as complex numbers.
When working with equations involving powers of \(i\), it’s essential to recognize the repeating pattern:

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

This cycle repeats, making computations involving higher powers of \(i\) manageable by reducing them to these fundamental values.

Quotient in Standard Form

A quotient in standard form for complex numbers means expressing the result as a complex number with a real part and an imaginary part, typically in the form of \(a + bi\). Standard form makes it easier to interpret and is a universally accepted way to express complex numbers. In the exercise example, you reverse-engineered the given complex division to a simpler form to determine the quotient by utilizing mathematical operations distinct to complex numbers.
In order to handle divisions involving complex numbers, especially when the denominator is not a standard real number, using its complex conjugate can simplify the expression. Multiply and divide by the conjugate, which transforms the imaginary component into a real one. Ultimately, simplifying using the rules of arithmetic results in a quotient that clearly articulates the real and imaginary components. Thus, achieving clarity in the expression of complex numbers.

Complex Conjugate

To simplify complex numbers, particularly when dividing, we often use a method involving the complex conjugate. The complex conjugate of a complex number \(a + bi\) is \(a - bi\). By multiplying both the numerator and the denominator of a complex fraction by the conjugate of the denominator, you eliminate the imaginary part from the denominator. This results in a denominator that is purely real, simplifying the division process.
In our specific exercise, the denominator \(-27i\) was simplified by multiplying with its conjugate, \(27i\):

• The result is \((-27i) \times (27i) = (27)^2 \times i^2 = 729 \times (-1) = -729\).
• This simplifies our calculation by removing the imaginary unit from the denominator.

Thus, the complex conjugate is a powerful tool to convert complex divisions into simpler arithmetic problems with real number denominators, allowing the division's outcome to be expressed in standard form.