Question

Find one or more values of each of the following complex expressions in the easiest way you can. \(\left(\frac{1+i \sqrt{3}}{\sqrt{2}+i \sqrt{2}}\right)^{50}\)

Short answer

\( \frac{\sqrt{3}}{2} + i \frac{1}{2} \)

Step-by-step solution

- Simplify the Expression

Start by simplifying the complex fraction \(\frac{1+i \, \sqrt{3}}{\sqrt{2}+i \, \sqrt{2}}\) using the polar form of complex numbers.

- Convert the Numerator and Denominator to Polar Form

Convert both the numerator (\(1+i \, \sqrt{3}\)) and the denominator (\(\sqrt{2} + i \, \sqrt{2}\)) to polar form. \(1 + i \, \sqrt{3} = 2 \, e^{i \, \frac{\pi}{3}}\) and \(\sqrt{2} + i \, \sqrt{2} = 2 \, e^{i \, \frac{\pi}{4}}\).

- Divide the Complex Numbers in Polar Form

Divide the polar forms: \(\frac{2 \, e^{i \, \frac{\pi}{3}}}{2 \, e^{i \, \frac{\pi}{4}}} = e^{i \, ( \frac{\pi}{3} - \frac{\pi}{4})} = e^{i \, \frac{\pi}{12}}\).

- Raise to the 50th Power

Raise the result to the power of 50: \(\left(e^{i \, \frac{\pi}{12}}\right)^{50} = e^{i \, \frac{50 \pi}{12}} = e^{i \, \frac{25 \pi}{6}}\).

- Simplify the Argument

Simplify the angle \(\frac{25 \pi}{6}\) to within \(\left[0, 2 \pi\right)\): \(\frac{25 \pi}{6} = 4 \pi + \frac{\pi}{6} = 2 \cdot 2 \pi + \frac{\pi}{6}\). Thus, it simplifies to \(e^{i \, \frac{\pi}{6}}\).

- Convert Back to Rectangular Form

Convert the polar form \(e^{i \, \frac{\pi}{6}}\) back to rectangular form: \(\cos \frac{\pi}{6} + i \, \sin \frac{\pi}{6} = \frac{\sqrt{3}}{2} + i \, \frac{1}{2}\).

Key concepts

Polar Form

Understanding the polar form of complex numbers is crucial for simplifying complex expressions. The polar form represents complex numbers in terms of their magnitude and angle.
This form is particularly useful because it simplifies the multiplication and division of complex numbers.
A complex number in polar form is written as \(r e^{i \theta}\) where \(r\) is the magnitude (distance from the origin) and \(\theta\) is the angle (argument) with the positive real axis.
To convert a complex number like \(1 + i \sqrt{3}\) to polar form:

• Find the magnitude: \(r = \sqrt{a^2 + b^2}\), here, \(r = \sqrt{1^2 + (\sqrt{3})^2} = 2\).
• Find the angle: \(\theta = \tan^{-1}(b/a)\), here \(\theta = \tan^{-1}(\sqrt{3}/1) = \frac{\pi}{3}\).
• Combine them: \(1 + i \sqrt{3} = 2 e^{i \frac{\pi}{3}}\).

Complex Fraction Simplification

When you encounter a complex fraction, simplifying it often involves converting the numerator and denominator into their polar forms.
For the fraction \(\frac{1 + i \sqrt{3}}{\sqrt{2} + i \sqrt{2}}\), start by finding the polar forms of both the numerator and the denominator.
The given step-by-step solution shows:

• Numerator: \(1 + i \sqrt{3} = 2 e^{i \frac{\pi}{3}}\).
• Denominator: \(\sqrt{2} + i \sqrt{2} = 2 e^{i \frac{\pi}{4}}\).

We then divide these forms: \(\frac{2 e^{i \frac{\pi}{3}}}{2 e^{i \frac{\pi}{4}}} = e^{i (\frac{\pi}{3} - \frac{\pi}{4})} = e^{i \frac{\pi}{12}}\).
This method simplifies the task significantly and prepares for raising the fraction to a power.

Euler's Formula

Euler's formula is a powerful tool in complex number arithmetic. It states that any complex number can be expressed as \(e^{i \theta} = \cos \theta + i \sin \theta\).
The step-by-step solution uses Euler's formula to convert between polar and rectangular forms.
For example, converting \(e^{i \frac{25\pi}{6}}\) back to rectangular form involves recognizing that\(e^{i \theta}\) represents \(\cos \theta + i \sin \theta\).
By simplifying \(\frac{25\pi}{6}\) to within \([0, 2\pi)\):

• \(\frac{25\pi}{6} - 4\pi = \frac{\pi}{6}\)

So \(e^{i \frac{25\pi}{6}}\) becomes \(e^{i \frac{\pi}{6}}\).
This can now be interpreted as \(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6} = \frac{\sqrt{3}}{2} + i \frac{1}{2}\).

Rectangular Form Conversion

Many problems require converting complex numbers back from polar to rectangular form. Euler's formula \(e^{i \theta} = \cos \theta + i \sin \theta\) makes this straightforward.
After simplifying the argument in the complex fraction solution, we arrive at \(e^{i \frac{\pi}{6}}\).

• Here, \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)
• And \(\sin \frac{\pi}{6} = \frac{1}{2}\)

Therefore, \(e^{i \frac{\pi}{6}}\) converts to the rectangular form \(\frac{\sqrt{3}}{2} + i \frac{1}{2}\).
This conversion is useful whenever you need the real and imaginary components of the complex number, especially for graphical representation or further arithmetic operations.