Problem 10 - Contemporary Precalculus

Question

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$(2+2 i)^{8}$$

Step-by-Step Solution

Answer

Answer: 256

Step 1: Convert the complex number to polar form

To convert the given complex number, \((2+2i)\), into polar form, we need to find its magnitude (r) and argument (θ). We obtain these by using the formulas: $$r = \sqrt{a^2 + b^2}$$ $$θ = \tan^{-1}\left(\frac{b}{a}\right)$$ Here, \(a=2\) and \(b=2\). Let's compute r and θ: $$r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$$ $$θ = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{π}{4}$$ So in polar form, the complex number is \((2\sqrt{2})(\cos(\frac{π}{4}) + i\sin(\frac{π}{4}))\).

Step 2: Use De Moivre's theorem to find the power

De Moivre's theorem states that for any complex number in polar form, \((r(\cosθ + i\sinθ))^n = r^n(\cos(nθ) + i\sin(nθ))\). In this exercise, we need to find the power of 8, so we apply De Moivre's theorem: $$(2\sqrt{2}(\cos(\frac{π}{4}) + i\sin(\frac{π}{4})))^8 = (2\sqrt{2})^8(\cos(8(\frac{π}{4})) + i\sin(8(\frac{π}{4})))$$ Now, compute \((2\sqrt{2})^8 = 256\) and \(8(\frac{π}{4})) = 2π\): $$256(\cos(2π) + i\sin(2π))$$

Step 3: Convert back to rectangular form

We can now convert this polar form back to rectangular form using the following formulas: $$a = r\cosθ$$ $$b = r\sinθ$$ Applying these formulas, we have: $$a = 256\cos(2π) = 256$$ $$b = 256\sin(2π) = 0$$ The complex number in rectangular form is \(256+0i\), or simply \(256\).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers, which combines a distance or magnitude (also called "radius") from the origin and an angle (also called "argument") to locate the point in the complex plane.
Here's a quick recap of how we convert a complex number like \(a + bi\) to its polar form:

• First, calculate the magnitude \(r\) using the formula: \(r = \sqrt{a^2 + b^2}\). For example, given \(2+2i\), the magnitude is \(\sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\).
• Second, determine the angle \(\theta\) using the formula: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Using the example, \(\tan^{-1}(\frac{2}{2}) = \frac{\pi}{4}\).

Thus, the polar form of the complex number \((2+2i)\) is \(2\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))\). This representation is useful in various mathematical operations, especially when dealing with powers and roots of complex numbers.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for raising complex numbers to a power once they are in polar form. It states: for a complex number \((r(\cos\theta + i\sin\theta))\) raised to any integer \(n\), the result is found by: \[ (r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta)) \]This simplifies the process considerably. Here's how it works in practice:

• Given \((2\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})))^8\), apply the theorem directly. Compute \((2\sqrt{2})^8 = 256\).
• For the angle, multiply by the power: \(8(\frac{\pi}{4}) = 2\pi\).

Thus, the simplified form using De Moivre's Theorem gives us \(256(\cos(2\pi) + i\sin(2\pi))\), turning a complex multiplication problem into a simple arithmetic calculation. It's a crucial theorem for working with complex powers.

Rectangular Form

In contrast to polar form, rectangular form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This is usually the form used for basic arithmetic operations on complex numbers.To convert from polar to rectangular form, you utilize:

• For the real part, compute \(a = r\cos\theta\).
• For the imaginary part, compute \(b = r\sin\theta\).

Given the polar result \(256(\cos(2\pi) + i\sin(2\pi))\), applying these formulas:

• Calculate: \(a = 256\cos(2\pi) = 256\), because \(\cos(2\pi) = 1\).
• Calculate: \(b = 256\sin(2\pi) = 0\), because \(\sin(2\pi) = 0\).

Thus, the rectangular form is simply \(256 + 0i\), or \(256\). This is usually the final form needed for most applications as it provides direct insight into the real and imaginary components.