Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-3 i)^{8}$$

Step-by-Step Solution

Answer

The eighth power of the complex number \(3-3i\) is \(117649 + 117649i\).

Step 1: Convert to Polar Form

First, the complex number \(3 - 3i\) is converted to polar form \(r(\cos\theta + i\sin\theta)\), where \(r = \sqrt{Re^2+Im^2}\) and \(\theta = \arctan\left(\frac{Im}{Re}\right)\). Therefore, the polar form of \(3 - 3i\) is thus \(3\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\)

Step 2: Apply DeMoivre's Theorem

According to DeMoivre's Theorem, \((r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i \sin(n\theta))\). Thus, \((3\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})))^8 = (3\sqrt{2})^8(\cos(-2\pi) + i \sin(-2\pi))\)

Step 3: Convert Result to Standard Form

Finally, convert the polar form back to the standard complex number form as it is asked in the task. The result from step 2 is \((3\sqrt{2})^8(\cos(-2\pi) + i \sin(-2\pi))\), which translates to \(117649 + 117649i\)

Key Concepts

Polar Form

To understand polar form, we must first appreciate that complex numbers can be represented not just on a number line, but also on a plane. This plane is often called the complex plane.
For instance, a complex number like \(3 - 3i\) can be thought of as a point or vector on this plane. In polar form, a complex number is represented as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude or modulus of the vector, and \(\theta\) is the angle, also known as the argument, with respect to the positive real axis.
Here’s how we find these values:

• **Magnitude \(r\)**: It is found using the formula \(r = \sqrt{Re^2+Im^2}\). For the number \(3 - 3i\), \(r = \sqrt{3^2 + (-3)^2} = 3\sqrt{2}\).
• **Argument \(\theta\)**: It is calculated as \(\theta = \arctan\left(\frac{Im}{Re}\right)\). For \(3 - 3i\), \(\theta = \arctan\left(\frac{-3}{3}\right) = -\frac{\pi}{4}\).

So, the polar form of \(3 - 3i\) is \(3\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\). Understanding the polar form is essential, especially when dealing with powers and roots of complex numbers.

Complex Numbers

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane using the horizontal axis for the real part and the vertical axis for the imaginary part. This form of numbers, written generally as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\), helps in solving equations that are otherwise unsolvable in the real number system.
The complex number \(3 - 3i\) can be split into:

• **Real part (\(a\))**: 3
• **Imaginary part (\(b\))**: -3, alongside the imaginary unit \(i\)

Understanding complex numbers provides a way to work with numbers that are outside the real scope and grasp concepts of geometry and mechanics. For instance, they are used in electrical engineering to represent voltages and currents.

Standard Form

The standard form of a complex number is simply its representation as \(a + bi\), where both \(a\) and \(b\) are real values that denote the real and imaginary parts, respectively.
Once we have worked with complex numbers in polar form, translating them back into standard form involves using trigonometric identities and finding the values for cosine and sine of the given angle, \(\theta\).

• In our case, after applying DeMoivre's Theorem to \((3\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})))^8\), we calculate the resultant expression to the power of 8 and convert it to standard form.
• The expression \((3\sqrt{2})^8(\cos(-2\pi) + i \sin(-2\pi))\) turns into \(117649 + 117649i\).

This shows how powerful translating between forms can be, simplifying computations and providing clear solutions, especially in algebra involving large powers or roots.