Problem 112 - 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. $$4(1-\sqrt{3} i)^{3}$$

Step-by-Step Solution

Answer

The indicated power of the complex number \(4(1-\sqrt{3} i)^{3}\) in standard form is \(-32 + 0i\).

Step 1. Convert complex number to polar form

First, convert the complex number \(1-\sqrt{3} i\) to polar form \(r(\cos \theta + i \sin \theta)\). The magnitude \(r\) is obtained by \(r = \sqrt{Re(z)^{2} + Im(z)^{2}}\), where \(Re(z)\) and \(Im(z)\) are the real and imaginary parts of the complex number respectively. The argument \(\theta\) (in radians) is found by \(\theta = \tan^{-1}( Im(z)/ Re(z) )\). Thus, \(r = \sqrt{ 1^{2} + (-\sqrt{3})^{2}} = 2\) and \(\theta = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}\). So, \(1-\sqrt{3} i = 2(\cos(-\frac{\pi}{3}) + i\sin(-\frac{\pi}{3}))\).

Step 2. Apply DeMoivre’s Theorem

Next, apply DeMoivre’s theorem to calculate the power of the complex number. DeMoivre's theorem states that \[(r(\cos \theta + i \sin \theta))^{n} = r^n (\cos n\theta + i \sin n\theta)\] Therefore, \((1-\sqrt{3} i)^{3} = (2(\cos(-\frac{\pi}{3}) + i\sin(-\frac{\pi}{3})))^{3} = 2^{3}(\cos(-\pi) + i\sin(-\pi)) = 8(-1 + 0i)\)

Step 3. Multiply by the constant

Now, multiply the result by the constant outside the parentheses. Here, the constant is 4. So, \(4(1-\sqrt{3} i)^{3} = 4 \times 8(-1 + 0i) = -32 + 0i\)

Key Concepts

Understanding Complex Numbers

Complex numbers are like enhanced real numbers. They are composed of two parts: a real part and an imaginary part. The imaginary part includes the imaginary unit, denoted as \( i \), which satisfies the equation \( i^2 = -1 \). A complex number usually has the form \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part.
Imagine them on a two-dimensional plane, called the complex plane, where the horizontal axis (x-axis) represents the real part and the vertical axis (y-axis) represents the imaginary part.
For example, in the complex number \(1-\sqrt{3}i\), \(1\) is the real part and \(-\sqrt{3}i\) is the imaginary part. This visualization helps in understanding operations like addition, subtraction, and even multiplication of complex numbers more intuitively.

Complex Numbers in Polar Form

The polar form of a complex number provides a different way of looking at it, using a magnitude and an angle, instead of the usual rectangular coordinates. In polar form, a complex number is expressed as \( r(\cos\theta + i\sin\theta) \), where \( r \) is the magnitude or modulus and \( \theta \) is the argument or angle.
How do we find these? The magnitude \( r \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \), similar to the Pythagorean theorem. In the example \(1-\sqrt{3}i\), the magnitude \( r = \sqrt{1^2 + (-\sqrt{3})^2} = 2 \).
The angle \( \theta \), typically in radians, is found using \( \tan^{-1}(b/a) \). For \(1 - \sqrt{3}i\), \( \theta = \tan^{-1} \left( \frac{-\sqrt{3}}{1} \right) = -\frac{\pi}{3} \). Thus, the polar form becomes \( 2(\cos(-\frac{\pi}{3}) + i\sin(-\frac{\pi}{3})) \).

Magnitude and Argument of Complex Numbers

The magnitude of a complex number tells us how "far" the number is from the origin on the complex plane, just like the length of a vector. It is always a non-negative number. The argument, on the other hand, is the "direction" of the complex number from the positive real axis, expressed in radians.
Let's highlight why these concepts are useful. By knowing the magnitude and argument, one can easily transform complex multiplication and power operations into simpler multiplication operations. This simplicity becomes evident with DeMoivre's Theorem, which leverages magnitude and argument for easy calculation of powers. For our example number \(1-\sqrt{3}i\), a magnitude of \(2\) and an argument of \(-\frac{\pi}{3}\) help us quickly compute its powers in polar form.

Powers of Complex Numbers using DeMoivre’s Theorem

When you need to calculate powers of complex numbers, DeMoivre’s Theorem is your hero. This theorem states that if a complex number is given in polar form as \( r(\cos\theta + i\sin\theta) \), then its \( n^{th} \) power can be found as \( r^n(\cos(n\theta) + i\sin(n\theta)) \).
It massively simplifies the process of raising complex numbers to a power. For our complex number \(1-\sqrt{3}i\), after converting it into polar form, DeMoivre's Theorem helps us quickly find \((1-\sqrt{3}i)^3\) by calculating \(2^3(\cos(-\pi) + i\sin(-\pi)) = -8 \).
After that, when multiplied by the constant \(4\), this becomes \(-32 + 0i\), bringing the solution in standard form. This technique is not just convenient, but also emphasizes the beauty and elegance of complex numbers.