Problem 80 - Precalculus Student Solutions Manual 5th

Question

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{5}{9 i}$$

Step-by-Step Solution

Answer

0 - \(\frac{5}{9}i\)

Step 1: Identify the problem

We are given the expression \(\frac{5}{9i}\) and need to convert it into the form \(a + bi\).

Step 2: Multiply by the conjugate

To eliminate the imaginary unit in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of \(9i\) is \(-i\), so we multiply by \(\frac{-i}{-i}\): \(\frac{5}{9i} \times \frac{-i}{-i}\)

Step 3: Simplify the expression

Multiply the numerators and the denominators.\(\frac{5 \times -i}{9i \times -i} = \frac{-5i}{-9i^2} \)

Step 4: Simplify further using properties of i

We know \(i^2 = -1\), so the denominator becomes \(-9(-1) = 9\): \(\frac{-5i}{9} = -\frac{5i}{9}\)

Step 5: Write the final answer in standard form

The expression in the form \(a + bi\) is \(0 - \frac{5}{9}i\).

Key Concepts

Standard Form a+bi

When dealing with complex numbers, we often want to express them in the standard form, which is written as \(a + bi\). Here, \(a\) is the real part, and \(bi\) is the imaginary part, where \(i\) is the imaginary unit with the property that \(i^2 = -1\). The goal of any operation involving complex numbers is usually to bring the result into this form.

In the problem given, the expression we started with was \(\frac{5}{9i}\). Initially, this doesn't look like the standard form \(a + bi\) because the imaginary unit \(i\) is in the denominator. Our task is to manipulate the expression until it conforms to the form \(a + bi\). This often involves some algebraic techniques such as rationalizing the denominator.

Multiplying by the Conjugate

To get rid of the imaginary unit in the denominator, we multiply by the conjugate. The conjugate of a purely imaginary number like \(9i\) is simply the negative of that number, which is \(-9i\). This technique is known as rationalizing the denominator.

The steps are:

• Original Expression: \(\frac{5}{9i}\)
• Identify the conjugate: \(-i\)
• Multiply: \(\frac{5}{9i} \times \frac{-i}{-i} = \frac{5(-i)}{9i(-i)}\)

By multiplying the numerator and denominator by the conjugate, we eliminate \(i\) from the denominator.

Simplifying Complex Expressions

After multiplying by the conjugate, we need to simplify the resulting expression:
1. First, multiply the numerators and the denominators: \(\frac{5 \times -i}{9i \times -i}\). This results in \(\frac{-5i}{-9i^2}\).
2. Recall that \(i^2 = -1\). So, the denominator \(-9i^2 = -9(-1) = 9\). The expression becomes \(\frac{-5i}{9}\).

Now, the result \(\frac{-5i}{9}\) can be written as \(0 - \frac{5}{9}i\) to match the standard form \(a + bi\), where \(a = 0\) and \(b = -\frac{5}{9}\). Hence, the final answer is \(0 - \frac{5}{9}i\).

The simplification process helps ensure that we manage complex expressions correctly and bring them into a consistent and useful format.