Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(-3-3 i)^{3}$$

Step-by-Step Solution

Answer

The power is 54 + 54i.

Step 1: Express in Polar Form

Convert the complex number \(-3-3i\) to its polar form. Recognize that the complex number can be represented as \(z = r(\cos\theta + i\sin\theta)\). The modulus \(r\) is calculated as \(r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}\). The argument \(\theta\) is \(\tan^{-1} \left(\frac{-3}{-3}\right) = \tan^{-1}(1) = \frac{3\pi}{4}\). Therefore, \(-3-3i = 3\sqrt{2}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})\).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, \((r(\cos\theta + i\sin\theta))^n = r^n (\cos(n\theta) + i\sin(n\theta))\). Substitute \(n=3\), \(r=3\sqrt{2}\), and \(\theta=\frac{3\pi}{4}\). Thus, \((-3-3i)^3 = (3\sqrt{2})^3 (\cos(3 \cdot \frac{3\pi}{4}) + i\sin(3 \cdot \frac{3\pi}{4}))\).

Step 3: Simplify Modulus

Calculate the modulus \(r^3\). Since \(r=3\sqrt{2}\), \(r^3 = (3\sqrt{2})^3 = 27 \cdot 2 \sqrt{2} = 54\sqrt{2}\).

Step 4: Calculate Angles

Calculate the angles: \(3\cdot\frac{3\pi}{4} = \frac{9\pi}{4}\). Simplify \(\frac{9\pi}{4}\) to a principal angle. Since \(\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}\), the principal angle is \(\frac{\pi}{4}\) (equivalent to 45 degrees).

Step 5: Convert Back to Rectangular Form

Use \(r^3\) and the principal angle to convert back to rectangular form: \((54\sqrt{2})(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\). Because \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), the rectangular form is: \(54\sqrt{2} \cdot \frac{\sqrt{2}}{2} + i \cdot 54\sqrt{2} \cdot \frac{\sqrt{2}}{2}\). Simplifying gives: \(54 + 54i\).

Key Concepts

Polar Form

Expressing complex numbers in polar form is a useful tool in many areas of mathematics, especially when dealing with powers and roots of complex numbers. A complex number given in rectangular form, such as \(-3 - 3i\), can be converted into polar form to simplify calculations. In polar form, a complex number is expressed as \(z = r(\cos \theta + i \sin \theta)\), where:

• \(r\) is the modulus of the complex number, calculated as \(r = \sqrt{a^2 + b^2}\)
• \(\theta\) is the argument, determined using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\)

In our example, the modulus of \(-3 - 3i\) is calculated as \(\sqrt{18} = 3\sqrt{2}\). The argument, located in the third quadrant because we consider the negative signs, is \(\frac{3\pi}{4}\). Thus, \(-3 - 3i\) converts to polar form as \(3\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\). Converting to polar form helps facilitate the application of certain theorems, such as De Moivre's Theorem.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for finding powers of complex numbers in polar form. It states that for any complex number \(z = r(\cos \theta + i \sin \theta)\), raised to a power \(n\), it can be expressed as: \[ (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]This theorem becomes very useful when dealing with complex number powers. Consider the complex number \(-3-3i\), which in polar form is \(3\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\). Applying De Moivre's Theorem for \(n=3\), we calculate as follows:

• The modulus becomes \(r^3 = (3\sqrt{2})^3 = 54\sqrt{2}\)
• The angle becomes \(3(\frac{3\pi}{4}) = \frac{9\pi}{4}\)
• We simplify the angle to a principal angle of \(\frac{\pi}{4}\)

This simplification allows other calculations to be easier, particularly when converting back to rectangular form.

Rectangular Form

Converting from polar back to rectangular form allows us to obtain an expression with real and imaginary components, typically easier to interpret. The process involves using the values of \(r^3\) and the principal angle to find the rectangular coordinates:Given the polar expression \(54\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\), find:

• \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)

Multiply \(54\sqrt{2}\) by each trigonometric component:

• Real part: \(54\sqrt{2} \times \frac{\sqrt{2}}{2} = 54\)
• Imaginary part: \(54\sqrt{2} \times \frac{\sqrt{2}}{2}i = 54i\)

So, the final expression in rectangular form is \(54 + 54i\). Converting between these forms helps bridge various mathematical processes, making manipulations with complex numbers more manageable.