Problem 75 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (2-2 i)^{8} $$

Step-by-Step Solution

Answer

The answer is 4096.

Step 1: Convert to Polar Form

The first step is to convert the complex number \(2 - 2i\) into polar form. The polar form is given by \(z = r(\cos\theta + i\sin\theta)\). First, find the modulus: \(r = \sqrt{a^2 + b^2} = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}\). Next, determine the argument \(\theta\) using \(\tan\theta = \frac{-2}{2} = -1\). Thus, \(\theta = \frac{7\pi}{4}\) since the complex number lies in the fourth quadrant.

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number \(z = r(\cos\theta + i\sin\theta)\), \(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\). Here, \(z = 2\sqrt{2}(\cos\frac{7\pi}{4} + i\sin \frac{7\pi}{4})\) and \(n = 8\). Therefore, \((2 - 2i)^8 = (2\sqrt{2})^8 \left(\cos(8\cdot \frac{7\pi}{4}) + i\sin(8\cdot \frac{7\pi}{4})\right)\).

Step 3: Calculate the Magnitude

Compute \((2\sqrt{2})^8 = 256^2\). The modulus is then \((2\sqrt{2})^8 = 4096\).

Step 4: Simplify the Argument

Calculate the angle: \(8 \cdot \frac{7\pi}{4} = 14\pi\), which is a full number of revolutions (since \(14\pi\) is equivalent to \(2\pi\cdot 7\)), meaning the resulting angle is \(0\) (since a whole number of full rotations brings you to the starting point on the circle).

Step 5: Compute the Power

Substitute the magnitude and simplified argument into the polar form equation: \[ (2\sqrt{2})^8 \left(\cos(0) + i\sin(0)\right) = 4096(1 + 0i) = 4096 \].

Step 6: Final Verification

Finally, verify the calculation by returning to the Cartesian form \(2 - 2i\) raised to \(8\), you get \(4096(1 + 0i)\), confirming that the power of this complex number indeed equals \(4096\).

Key Concepts

Complex Numbers

Understanding complex numbers is like learning a new language in mathematics. A complex number consists of two parts: a real part and an imaginary part. We write them in the form \(a + bi\) where \(a\) is the real part, and \(b\) is the imaginary part, with \(i\) being the imaginary unit, which satisfies \(i^2 = -1\).

Complex numbers are useful in various fields like engineering, physics, and computer science because they can represent two-dimensional numbers. In our problem, the complex number is \(2 - 2i\), where \(a = 2\) and \(b = -2\). This representation allows for operations like addition, subtraction, and particularly crucially for this exercise—multiplication and exponentiation with De Moivre's Theorem.

• Real Part (a): Determines the horizontal placement on the complex plane.
• Imaginary Part (b): Determines the vertical placement on the complex plane.

Polar Form

To solve problems involving the power of a complex number, it is often more convenient to express the complex number in polar form. The polar form presents a complex number in terms of its modulus and argument.

For a complex number \( z = a + bi \), the polar form is \( z = r(\cos \theta + i\sin \theta) \). Here, \( r \) is the modulus (or magnitude), and \( \theta \) is the argument (or angle). Switching to this form simplifies the process of exponentiation, especially when applying De Moivre's Theorem.

• Modulus (r): The distance of the complex number from the origin.
• Argument (θ): The angle formed with the positive real axis.

Modulus and Argument

The modulus and argument are key components of a complex number's polar form. The modulus \(r\) of \(2 - 2i\) is computed as \(r = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}\), which represents the distance from the origin to the point \((2, -2)\) on the complex plane.

The argument \( \theta \) indicates the counterclockwise angle from the positive real axis to the line representing the complex number. For our number, \(\tan \theta = \frac{-2}{2} = -1\), which gives \(\theta = \frac{7\pi}{4}\), placing \(2 - 2i\) in the fourth quadrant.

• Conversion to Rectangular Form: \( x = r \cos \theta,~ y = r \sin \theta \)
• Quadrant Consideration: Determines the appropriate angle \(\theta\) based on signs of \(a\) and \(b\).

Mathematical Proof

Applying mathematical proofs in complex numbers, particularly with De Moivre's Theorem, validates the conversions and operations we perform. De Moivre's Theorem is a powerful statement used in mathematics to compute powers of complex numbers in polar form.

The theorem states that if \(z = r(\cos \theta + i\sin \theta)\), then \(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\). This simplifies the raising of complex numbers to powers, turning multiplication and exponential functions into mere calculations of \(r^n\) and multiple angles. For \( (2 - 2i)^8 \), using De Moivre's Theorem makes it straightforward to compute exponentiation without going through more complex algebraic manipulation.

• Verification: Cross-check initial results by repeating calculations or converting back to false check discrepancies.
• Angle Simplification: Use full-circle equivalences to bring angles into standard position.

Exponentiation of Complex Numbers

Exponentiating complex numbers directly in their rectangular form is usually cumbersome, which is where De Moivre's Theorem shines as a valuable tool. By converting complex numbers into their polar form, we can easily perform exponentiation through De Moivre's Theorem.

For our problem, the conversion allowed for raising \(2 - 2i\) to the power of 8 into a simple calculation: finding \((2\sqrt{2})^8\) and calculating the argument over the circle's revolutions. The result of 4096 shows the power of using polar coordinates and simplifies a potentially intensive computation.

• Efficient Computation: Reduces the error-prone nature of manual multiplication.
• Cycle Utilization: Leveraging trigonometric function periodicity to simplify calculations.