Problem 71 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (2 \sqrt{3}+2 i)^{5} $$

Step-by-Step Solution

Answer

The result is \(-512\sqrt{3} + 512i\).

Step 1: Convert to Polar Form

First, express the complex number in polar form. The complex number given is \(z = 2\sqrt{3} + 2i\). To do this, compute the modulus \(r\) and the argument \(\theta\). The modulus is given by \(r = \sqrt{(2\sqrt{3})^2 + 2^2} = \sqrt{12 + 4} = \sqrt{16} = 4\). The argument \(\theta\) is \( \theta = \tan^{-1}\left(\frac{2}{2\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\). The polar form is \(4(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that \((r (\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))\). Substitute \(r = 4\), \(\theta = \frac{\pi}{6}\), and \(n = 5\): \( (4 (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}))^{5} = 4^5 (\cos(5 \times \frac{\pi}{6}) + i \sin(5 \times \frac{\pi}{6}))\).

Step 3: Compute Powers and Angles

Calculate \(4^5\) which is 1024. Compute the new angle, \(5 \times \frac{\pi}{6} = \frac{5\pi}{6}\). So the expression becomes \(1024 (\cos(\frac{5\pi}{6}) + i \sin(\frac{5\pi}{6}))\).

Step 4: Simplify Trigonometric Expressions

Use trigonometric values: \(\cos(\frac{5\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}\) and \(\sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}\). Substitute these back: \(1024 \left(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\right)\).

Step 5: Final Answer in Rectangular Form

Multiply out the terms: \(1024 \left(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\right) = 1024 \times -\frac{\sqrt{3}}{2} + 1024 \times i\frac{1}{2}\). Thus the rectangular form is \(-512\sqrt{3} + 512i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).
Complex numbers allow us to perform calculations in a two-dimensional space. For instance, the complex number \(2\sqrt{3} + 2i\) combines a real part \(2\sqrt{3}\) and an imaginary part \(2i\).

• Real part: Represents the horizontal component.
• Imaginary part: Represents the vertical component.

Understanding how complex numbers work is essential in advanced mathematics, especially in fields like electrical engineering and quantum physics.

Polar Form

The polar form of a complex number is another way to express the number, based on its magnitude and direction rather than its horizontal and vertical components. While the rectangular form is \(a + bi\), the polar form uses the modulus and the argument: \(r(\cos \theta + i \sin \theta)\).
Here, \(r\) is the distance from the origin in the complex plane, and \(\theta\) is the angle from the positive x-axis. For the complex number \(2\sqrt{3} + 2i\), we converted to polar form with:

• Modulus \(r = 4\)
• Argument \(\theta = \frac{\pi}{6}\)

Thus, the polar form becomes \(4(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\). This form is particularly useful in computing powers and roots of complex numbers.

Trigonometric Form

The trigonometric form of complex numbers, often used interchangeably with polar form, allows for expressing a complex number in terms of trigonometric functions. It's written as \(r(\cos \theta + i \sin \theta)\) and is helpful in visualizing the number's properties in the complex plane.
The cosine function corresponds to the horizontal component (real part), and the sine function corresponds to the vertical component (imaginary part). This form is very useful in multiplying and finding powers of complex numbers - tasks made simpler compared to handling the binomial nature of rectangular form.
For example, the trigonometric representation for \(2\sqrt{3} + 2i\) supports easier calculations when raised to any power, thanks to De Moivre's Theorem.

Modulus and Argument

The modulus and argument are crucial concepts in understanding complex numbers. The modulus, often denoted by \(r\), represents the distance of the complex number from the origin in the complex plane, calculated by \(r = \sqrt{a^2 + b^2}\). For \(2\sqrt{3} + 2i\), the modulus is \(4\).
The argument, \(\theta\), is the angle made with the positive x-axis, calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). For our complex number, the argument is \(\frac{\pi}{6}\). These two components together fully describe the position of a complex number in the complex plane and serve as the foundation for its polar form.
Understanding these fundamental aspects allows for better application of concepts like De Moivre's Theorem, simplifying operations with complex numbers.

Powers of Complex Numbers

Raising complex numbers to a power can be quite challenging in their rectangular form. This is where De Moivre's Theorem comes into play, greatly simplifying the process using the polar or trigonometric form.
De Moivre's Theorem states that for a complex number in polar form \((r(\cos \theta + i \sin \theta))^n\), the power is \(r^n (\cos(n\theta) + i \sin(n\theta))\).
This makes it accessible to compute high powers of complex numbers efficiently. For our example \((2\sqrt{3} + 2i)^5\), once converted to polar form, we applied the theorem to find the power as \((1024)(\cos(\frac{5\pi}{6}) + i\sin(\frac{5\pi}{6}))\).
The final step is converting back into rectangular form for a comprehensive solution.