Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(2-2 i \sqrt{3})^{4}$$

Step-by-Step Solution

Answer

The power is \(-32 + 192i\sqrt{3}\) in rectangular form.

Step 1: Identify the Complex Number

The complex number given is \(2 - 2i \sqrt{3}\), which is in the form \(a + bi\), where \(a = 2\) and \(b = -2 \sqrt{3}\).

Step 2: Apply Binomial Theorem

We will expand \((2 - 2i\sqrt{3})^4\) using the binomial theorem, which states \[(a + b)^n = \sum_{k=0}^{n} {\binom{n}{k} a^{n-k} b^k}\]In this case, \(a = 2\), \(b = -2i\sqrt{3}\), and \(n = 4\).

Step 3: Calculate Each Term

We need to calculate each term in the expansion. The general term is \[\binom{4}{k} (2)^{4-k} (-2i\sqrt{3})^k\]- For \(k = 0\): \(\binom{4}{0}(2)^4(-2i \sqrt{3})^0 = 16\)- For \(k = 1\): \(\binom{4}{1}(2)^3(-2i\sqrt{3}) = -64i\sqrt{3}\)- For \(k = 2\): \(\binom{4}{2}(2)^2(-2i\sqrt{3})^2 = -192\)- For \(k = 3\): \(\binom{4}{3}(2)(-2i\sqrt{3})^3 = 256i \sqrt{3}\)- For \(k = 4\): \(\binom{4}{4}(-2i\sqrt{3})^4 = 144\)

Step 4: Add All Terms Together

Add the real parts and the imaginary parts separately from the expansion results: Real part: \(16 - 192 + 144 = -32\)Imaginary part: \(-64i\sqrt{3} + 256i\sqrt{3} = 192i\sqrt{3}\)Thus the expression \((2 - 2i \sqrt{3})^4\) simplifies to \(-32 + 192i\sqrt{3}\).

Key Concepts

Rectangular Form

In the world of complex numbers, the rectangular form is a common way of representing these numbers. Each complex number can be expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Here, \(i\) represents the imaginary unit, which satisfies \(i^2 = -1\).

When solving problems involving complex numbers, it's crucial to write the final answer in rectangular form. This makes it easier to distinguish the real and imaginary components.

• For the given expression \((2 - 2i\sqrt{3})^4\), we first identify the complex number: \(a = 2\) and \(b = -2\sqrt{3}\).
• After computing the power, as shown in the step-by-step solution, we end up with \(-32 + 192i\sqrt{3}\), which clearly displays the real and imaginary parts separately.

Understanding rectangular form helps in visualizing complex numbers on the complex plane, with the real part along the x-axis and the imaginary part along the y-axis.

Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions that are raised to a power. It states:\[(a + b)^n = \sum_{k=0}^{n} {\binom{n}{k} a^{n-k} b^k}\]This means you can express \((a + b)^n\) as a sum of terms, where each term comes from different combinations of \(a\) and \(b\).

In our exercise, we apply the binomial theorem to \((2 - 2i\sqrt{3})^4\). Here:

• \(a = 2\), \(b = -2i\sqrt{3}\), and \(n = 4\).
• For each value of \(k\) from 0 to 4, calculate the terms \(\binom{4}{k} (2)^{4-k} (-2i\sqrt{3})^k\).

Each step involves multiplying the terms using the combinations given by \(\binom{n}{k}\) to construct an expanded form, ultimately leading to a straightforward expression in rectangular form. This theorem provides a systematic way to tackle powers of sums, especially useful when imaginary numbers are involved.

Imaginary Numbers

Imaginary numbers often seem mysterious at first, but they have a very simple premise. They are numbers that include \(i\), the imaginary unit, which satisfies \(i^2 = -1\). Imaginary numbers are extremely helpful when working with square roots of negative numbers, helping extend our number system.

Within complex numbers, the imaginary part is always accompanied by \(i\). For example, in the complex number \(-2i\sqrt{3}\), \(-2\sqrt{3}\) is the coefficient of the imaginary part.

• In our exercise, we use this concept when identifying the complex number \(2 - 2i\sqrt{3}\).
• The calculations in the binomial expansion consistently deal with products of \(i\), illustrating how imaginary numbers interact with real numbers.

Grasping imaginary numbers enhances understanding of complex numbers as a whole, allowing their use in calculations that require handling non-real quantities.

Complex Number Powers

Finding powers of complex numbers can seem daunting, but by breaking down the complex number into its components, the task becomes manageable. Working in rectangular form, one can use the binomial theorem to expand powers efficiently.

The power in our exercise \((2 - 2i\sqrt{3})^4\) demonstrates how:

• The binomial theorem helps systematically expand each term with respect to its power.
• Dealing with \(i\) is made easy by remembering that \(i^2 = -1\).

By calculating and adding up all the terms, the final result simplifies into an easily understood rectangular form.

Whether in mathematics, physics, or engineering, the ability to handle complex number powers is invaluable. It extends the functionality of complex numbers to more advanced concepts and applications across various fields.