91Ó°ÊÓ

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.

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.

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.