Problem 81 - Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i\) $$ (6-8 i)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-28 - 96i\).

Step 1: Identify the Expression

We are given the expression \((6 - 8i)^2 \)which needs to be simplified and written in the form \(a + bi\).

Step 2: Apply the Formula for Squaring a Binomial

Recall the formula \((a-b)^2 = a^2 - 2ab + b^2\).Apply it here: \((6 - 8i)^2 = 6^2 - 2(6)(8i) + (8i)^2\).

Step 3: Calculate Each Term

Compute each term separately:- \(6^2 = 36\)- \(-2(6)(8i) = -96i\)- \((8i)^2 = 64i^2\). Remember that \(i^2 = -1\), so \((8i)^2 = 64(-1) = -64\).

Step 4: Simplify the Expression

Substitute the calculated terms back into the expression:\(36 - 96i - 64\). Combine like terms: the real parts \(36 - 64 = -28\) and the imaginary part \(-96i\) stays the same.

Step 5: Write the Answer in Standard Form

The simplified form of the expression is:\(-28 - 96i\). This is already in the form \(a + bi\), where \(a = -28\) and \(b = -96\).

Key Concepts

Binomial Squaring

Binomial squaring is a fundamental mathematical technique often used for expanding expressions of the form \((a-b)^2\) or \((a+b)^2\). Understanding this is crucial when working with complex numbers. The general formula for squaring a binomial is given by:

• \((a-b)^2 = a^2 - 2ab + b^2\)
• \((a+b)^2 = a^2 + 2ab + b^2\)

This formula helps in breaking down and simplifying expressions by expanding each part individually. Let's apply this to solve and transform complex expressions such as \((6 - 8i)^2\). Begin by calculating each component individually:- The square of the first term: \(6^2 = 36\).- The product of both terms, doubled: \(-2 \times 6 \times 8i = -96i\).- The square of the second term: \((8i)^2\). After computing these, gather the results. This helps keep track of each element, avoiding confusion in simplifying complex expressions.

Imaginary Unit

The imaginary unit, commonly denoted as \(i\), represents the square root of \(-1\). This is a cornerstone concept in complex numbers, making it possible to extend the real number system. This magical unit helps solve equations that don't have solutions within the realm of real numbers.Here are a few properties to remember:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Understanding these properties aids in simplifying expressions involving complex numbers. For example, in the expression \((8i)^2\), we use the property \(i^2 = -1\), so \((8i)^2 = 64i^2 = 64 \times (-1) = -64\). Thus, efficiently managing the imaginary unit is key in tasks such as expanding and simplifying expressions.

Simplifying Expressions

Simplifying expressions, especially those involving complex numbers, requires special care to combine like terms correctly. Begin with calculating the numerical values for real and imaginary parts. Here's how you can methodically simplify them:1. **Compute Each Component**: Each part of a binomial is calculated separately. - For real numbers, compute using basic arithmetic rules. - Use the imaginary unit properties for imaginary numbers.2. **Combine Like Terms**: Gather all real numbers and combine them. Next, do the same for imaginary numbers. This often involves algebraically adding or subtracting these components.For instance, with the expression \((6 - 8i)^2\), once expanded and calculated, it becomes \(36 - 96i - 64\). Following your mathematical operations, combine the real parts, resulting in \(36 - 64 = -28\), and then process the imaginary part, \(-96i\).Finally, ensure the simplified expression is in the standard form \(a + bi\), which in this case is \(-28 - 96i\). Grasping these steps guarantees that your results are both accurate and easy to interpret.