Problem 68 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-2 i)^{8}$$

Step-by-Step Solution

Answer

The 8th power of the complex number \((3 - 2i)\) in standard form is calculated using DeMoivre's theorem, followed by converting the result back to standard form.

Step 1: Convert to polar form

The polar form of a complex number is given by \(r (\cos \theta+ i \sin \theta)\), where \(r=\sqrt{(a^2+b^2)}\) and \(\theta=\arctan(\frac{b}{a})\). Here, \(a=3\) and \(b=-2\). When calculating, this results in \(r=\sqrt{(3^2+(-2)^2)}\) and \(\theta=\arctan(\frac{-2}{3})\). So, \(r = \sqrt{13}\) and \(\theta = \arctan(\frac{-2}{3})\). Put these values into the formula to convert to polar form.

Step 2: Apply DeMoivre's theorem

Now, apply DeMoivre's theorem: \((r (\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\). Here, \(n=8\). So, raising the number to the power 8 gives \(\sqrt{13}^8 (\cos 8\arctan(\frac{-2}{3}) + i \sin 8\arctan(\frac{-2}{3}))\).

Step 3: Convert back to standard form

The final part of the problem involves converting this polar form back to the standard form \(a+bi\). This requires bringing the \(cos\) and \(sin\) terms back into their equivalent form. The result is the final power of the complex number in standard form.

Key Concepts

Complex Numbers

Complex numbers might seem complicated initially, but they are manageable with practice. Essentially, a complex number is a number that comprises two parts:

• A real component represented by \(a\)
• An imaginary component represented by \(bi\)

The imaginary unit, \(i\), is defined as the square root of -1, which means \(i^2 = -1\). A typical complex number is expressed as \(a + bi\). For example, in the exercise, we have the complex number \(3 - 2i\), where 3 is the real part and -2i is the imaginary part.
Complex numbers allow us to perform arithmetic operations and solve equations that may not be possible with just real numbers. They are essential in various mathematical and engineering applications. The beauty of complex numbers lies in their ability to represent rotations and oscillations, making them invaluable in physics and electrical engineering.

Polar Form

When working with complex numbers, the polar form is another way to express them. It becomes extremely useful, especially when dealing with powers of complex numbers, as seen in DeMoivre's Theorem.

• **Magnitude \(r\):** This is the length of the vector representation of the complex number and is calculated using \(r = \sqrt{a^2 + b^2}\).
• **Argument \(\theta\):** This is the angle the vector makes with the positive real axis. It's obtained using \(\theta = \arctan(\frac{b}{a})\).

For the complex number \(3 - 2i\), the magnitude \(r\) is found by\(\sqrt{3^2 + (-2)^2} = \sqrt{13}\), and the argument \(\theta = \arctan(\frac{-2}{3})\). Therefore, the polar form of \(3 - 2i\) can be expressed as \(\sqrt{13}(\cos(\theta) + i\sin(\theta))\).
The polar form is especially beneficial when multiplying or powering complex numbers because it simplifies the operation to multiplying magnitudes and adding arguments.

Standard Form

The final aim when dealing with a complex number in a problem is to express it in "standard form," which is the basic form \(a + bi\). After applying DeMoivre's Theorem to find powers in polar form, converting back to standard form is crucial for interpretation and understanding of the results.
To do so, you will calculate the cosine and sine of the multiple angles involved and multiply them by the magnitude part. This transforms your polar form back into a more familiar arrangement. For example, the result is a computation involving both the actual parts \(\cos(n\theta)\) and \(\sin(n\theta)\) after applying the necessary operations from DeMoivre's theorem.
Ultimately, standard form lets you clearly see what the real and imaginary parts of your powered complex number are, which can then be used in further applications or evaluations.