Problem 113 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ (4-8 i)^{2} $$

Step-by-Step Solution

Answer

The answer is \(-48 - 64i\).

Step 1: Expand the Square

The expression \((4-8i)^2\) is a binomial square. We apply the formula for expanding \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4\) and \(b = 8i\).

Step 2: Calculate Each Term

Calculate each term separately: \(a^2 = 4^2 = 16\), \(-2ab = -2(4)(8i) = -64i\), and \((8i)^2 = 64i^2\).

Step 3: Simplify the Imaginary Part

Since \(i^2 = -1\), substitute to find \(64i^2 = 64(-1) = -64\).

Step 4: Combine Real and Imaginary Parts

Combine all terms: \(16 - 64 + (-64i) = -48 - 64i\).

Step 5: Write in Standard Form

The result is written in the form \(a + bi\) as \(-48 - 64i\).

Key Concepts

Binomial Expansion

The binomial expansion technique is very useful when dealing with squared expressions like \((4 - 8i)^2\). The key is to apply the formula for expanding binomials, \((a - b)^2 = a^2 - 2ab + b^2\). This formula helps break down the binomial into simpler parts.
The process involves calculating each term separately:

• First, you square the first term, which is \(a^2\).
• Then, double the product of both terms, which is \(-2ab\).
• Finally, square the second term, which is \(b^2\). In our equation, this involves the imaginary number \(i\).

Once these steps are completed, you can easily combine them to simplify the expression. Understanding this process also aids in handling more complex operations involving binomials.

Imaginary Unit

Imaginary numbers may seem complex at first, but they are quite straightforward once you grasp the basics. The imaginary unit is denoted by \(i\) and is defined as the square root of \(-1\), which means that \(i^2 = -1\). This unique property is what makes imaginary numbers both intriguing and formidable.
In the provided exercise, this property was used during the expansion process. For instance, when we calculate \((8i)^2\), it simplifies to \(64i^2\). Given that \(i^2 = -1\), you substitute to get \(64(-1) = -64\).

• Remember: whenever you see \(i^2\), it should be replaced with \(-1\).

This substitution not only simplifies complex expressions but also reveals the real-life applications of imaginary numbers in fields like engineering and physics.

Standard Form of Complex Numbers

Complex numbers are typically composed of a real part and an imaginary part, and are expressed in the standard form \(a + bi\), where:

• \(a\) is the real part of the complex number.
• \(b\) is the coefficient of the imaginary part represented by \(i\).

For the given problem, after performing the calculations, we combine the individual components—real numbers and imaginary numbers—to arrive at the result. With our example, combining terms resulted in \(-48 - 64i\). This expression is already in the standard form of a complex number.
The standard form makes it easier to understand and communicate the composition of complex numbers. This format also facilitates operations like addition, subtraction, and comparison with other complex numbers. Mastering this representation is crucial for further exploration of complex operations and their applications.