Problem 80 - Precalculus

Question

Calculate \(\frac{\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{14}}{\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{20}}\).

Step-by-Step Solution

Answer

The value of the expression is \( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \).

Step 1: Recognize Complex Numbers in Polar Form

The complex number given, \( \frac{1}{2} + \frac{\sqrt{3}}{2}i \), is in rectangular form. Notice this complex number has a magnitude of 1, as \( \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \). It can be rewritten in polar form as \( e^{i\theta} \), where \( \theta = 60^\circ = \frac{\pi}{3} \).

Step 2: Conversion of Second Complex Number

Similarly, the complex number \( \frac{1}{2} - \frac{\sqrt{3}}{2}i \) also has a magnitude of 1, and it converts to polar form as \( e^{-i\theta} \), where \( \theta = 60^\circ = \frac{\pi}{3} \).

Step 3: Apply Polar Form to Exponents

Express the powers using Euler's formula: \( \left(e^{i\frac{\pi}{3}}\right)^{14} \) and \( \left(e^{-i\frac{\pi}{3}}\right)^{20} \). Simplify these to \( e^{i \cdot 14 \cdot \frac{\pi}{3}} \) and \( e^{-i \cdot 20 \cdot \frac{\pi}{3}} \).

Step 4: Simplify the Exponents

Calculate the exponents: \( 14 \times \frac{\pi}{3} = \frac{14\pi}{3} \) and \( 20 \times -\frac{\pi}{3} = -\frac{20\pi}{3} \). Now, calculate the overall exponent by subtracting: \( \frac{14\pi}{3} - (-\frac{20\pi}{3}) = \frac{14\pi}{3} + \frac{20\pi}{3} = \frac{34\pi}{3} \).

Step 5: Simplify Using Periodicity of the Complex Exponential Function

Since the exponential function is periodic with period \(2\pi\), calculate the equivalent angle within \([0, 2\pi)\): \(\frac{34\pi}{3} = 2\pi \times 5 + \frac{4\pi}{3}\). Therefore, the function simplifies to \( e^{i \frac{4\pi}{3}} \).

Step 6: Convert Back to Rectangular Form

Recall that \( e^{i \frac{4\pi}{3}} \) corresponds to an angle of \( 240^\circ \), which equals \( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \) in rectangular form. Thus, the value of the expression is \( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \).

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers. Instead of using the usual rectangular form, such as \( a + bi \), polar form represents complex numbers in terms of magnitude and angle. This form is particularly useful for multiplication, division, and exponentiation of complex numbers.
To convert a complex number to polar form, follow these simple steps:

• Calculate the magnitude \( r \) of the complex number. For a complex number \( a + bi \), the magnitude is \( r = \sqrt{a^2 + b^2} \).


• Determine the angle \( \theta \), which is measured from the positive real axis. If \( a + bi \) is in the first quadrant, \( \theta = \tan^{-1}(\frac{b}{a}) \).

Once you have \( r \) and \( \theta \), the polar form is expressed as \( r(\cos \theta + i\sin \theta) \). Alternatively, using Euler's formula, it can also be expressed as \( re^{i\theta} \). This conversion simplifies the calculations of powers and roots of complex numbers.

Euler's Formula

Euler's Formula is a fascinating bridge between complex numbers and exponential functions. It is expressed as:

• \( e^{i\theta} = \cos\theta + i\sin\theta \)

Here, \( \theta \) represents the angle in radians. This formula is highly useful because it simplifies many calculations involving complex numbers. For instance, complex exponents and logarithms become straightforward when using Euler's formula.
If you have a complex number in polar form \( re^{i\theta} \), and you raise it to an exponent, you can easily calculate it using:
\( (re^{i\theta})^n = r^n e^{i n \theta} \)

• This makes computations much easier, particularly in this exercise, where raising complex numbers to powers is involved.
• It’s a key tool in simplifying complex calculations and is an essential part of electrical engineering, physics, and other fields where wave motion or periodic phenomena are important.

Rectangular Form

Rectangular form, also known as Cartesian form, represents complex numbers as \( a + bi \). Here, \( a \) and \( b \) are real numbers; \( a \) is the real part, and \( b \) is the imaginary part.
Converting a polar form complex number back to rectangular form involves simple trigonometric calculations.
For a complex number in polar form \( re^{i\theta} \):

• The real part \( a = r \cos\theta \)


• The imaginary part \( b = r \sin\theta \)

In this exercise, converting from polar to rectangular form allows the final answer \( e^{i\frac{4\pi}{3}} \) to be rewritten as \( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \).
Rectangular form is often preferred for plotting points on the complex plane or for performing addition and subtraction with complex numbers.