91Ó°ÊÓ

A complex number can be expressed in different forms, and one of the most useful forms is the polar form. To convert a complex number like \(9 + 9i\) into polar form, we need to find its magnitude and argument.
The magnitude \(r\) is calculated as the distance from the origin in a complex plane. For our example, the magnitude is \(r = \sqrt{9^2 + 9^2} = \sqrt{162} = 9\sqrt{2}\).

The angle or argument \(\theta\) is the angle the line connecting the origin and the point \((9, 9)\) makes with the positive real axis. This angle is found using \(\tan^{-1}\left(\frac{9}{9}\right) = \frac{\pi}{4}\).
Thus, the polar form of \(9 + 9i\) is \(9\sqrt{2} \text{cis} \frac{\pi}{4}\). Using polar form can simplify operations like multiplication and exponentiation. This is particularly useful when used alongside De Moivre's Theorem. By expressing a complex number in polar form, it becomes much easier to take powers and roots.

De Moivre's Theorem is a powerful tool for dealing with powers of complex numbers expressed in polar form. It states that for any complex number in polar form \(r \text{cis} \theta\), raising it to the power of \(n\) is achieved by \(r^n \text{cis}(n\theta)\).

Let's apply this theorem to \( (9+9i)^6 \). We've already converted \(9 + 9i\) into polar form as \(9\sqrt{2} \text{cis} \frac{\pi}{4}\). Now, using De Moivre's Theorem, we calculate:

• Raise the magnitude to the sixth power: \((9\sqrt{2})^6 = 46656\).
• Multiply the angle by 6: \(6 \times \frac{\pi}{4} = \frac{3\pi}{2}\).

Thus, \((9 + 9i)^6 = 46656 \text{cis} \frac{3\pi}{2}\).
De Moivre's Theorem makes the process of exponentiation straightforward by transforming it into simple multiplication and exponentiation, without dealing with complex expansion directly.

The rectangular form of a complex number is the standard representation as \(a + bi\), where \(a\) and \(b\) are real numbers. After applying operations in polar form, it’s often necessary to convert back to rectangular form for intuitive understanding or applicability.

In this exercise, we found \((9+9i)^6 = 46656 \text{cis} \frac{3\pi}{2}\).
To convert this back to rectangular form, we utilize the angle \(\frac{3\pi}{2}\), which corresponds to a point on the unit circle at \((0, -1)\).

• The real part is \(46656 \times 0 = 0\).
• The imaginary part is \(46656 \times (-1) = -46656\).

Thus, the rectangular form is \(-46656i\).
The transformation between rectangular and polar forms allows for seamless transition between different operations on complex numbers, leveraging the strengths of both representations.