Problem 35 - Elementary Algebra

Question

Perform the operations. $$ (12+23 i)(13-12 i) $$

Step-by-Step Solution

Answer

The product is \(432 + 155i\).

Step 1: Understand the Expression

Given is the multiplication of two complex numbers: \((12+23i)\) and \((13-12i)\). We need to multiply these two complex numbers together.

Step 2: Use the Distributive Property

We'll apply the distributive property (also known as FOIL for binomials) to multiply these two complex numbers:\[(12+23i)(13-12i) = 12 \cdot 13 + 12 \cdot (-12i) + 23i \cdot 13 + 23i \cdot (-12i)\]

Step 3: Calculate Each Term

Compute each term separately:1. \(12 \cdot 13 = 156\)2. \(12 \cdot (-12i) = -144i\)3. \(23i \cdot 13 = 299i\)4. \(23i \cdot (-12i) = -276i^2\)

Step 4: Simplify Using \(i^2 = -1\)

Since \(i^2 = -1\), we replace \(-276i^2\) with \(276\):\[-276i^2 = -276(-1) = 276\]

Step 5: Combine Like Terms

Combine the real and imaginary parts:- Real: \(156 + 276 = 432\)- Imaginary: \(-144i + 299i = 155i\)

Step 6: Write the Final Result

The result of multiplying \((12+23i)(13-12i)\) is \(432 + 155i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions and solve equations. When multiplying two binomials like \((12+23i)\) and \((13-12i)\), we can think of this as distributing each term in one binomial across every term in the other binomial. This process ensures that each combination of terms is accounted for, resulting in four distinct products when dealing with complex numbers.Here’s how it works in our example:

• Multiply the first terms: \(12 \cdot 13 = 156\)
• Multiply the outer terms: \(12 \cdot (-12i) = -144i\)
• Multiply the inner terms: \(23i \cdot 13 = 299i\)
• Multiply the last terms: \(23i \cdot (-12i) = -276i^2\)

Each of these products contributes to the final expression, which can then be combined by adding these terms together.

Binomials Multiplication

Multiplying binomials is a particular case of using the distributive property.In expressions like \((a + bi)(c + di)\),we apply the distributive property to expand these products. This method is often summarized by the acronym FOIL, which stands for First, Outer, Inner, and Last.For the problem at hand, we apply FOIL to determine:

• First (F): \(12 \times 13 = 156\)
• Outer (O): \(12 \times -12i = -144i\)
• Inner (I): \(23i \times 13 = 299i\)
• Last (L): \(23i \times -12i = -276i^2\)

When expanding binomials that include imaginary numbers, we're not just adding and multiplying as in typical algebra – we're also incorporating the imaginary unit \(i\) as part of the calculations, which will impact the final result.

Imaginary Unit

The imaginary unit, denoted as \(i\), is crucial in complex number arithmetic. It is defined such that \(i^2 = -1\). This property is what distinguishes imaginary numbers from real numbers and allows for unique calculations in multiplication.In our exercise, one key step involves handling the term \(-276i^2\). By recalling that \(i^2 = -1\), we can simplify this to \[-276(-1) = 276\].Thus, the expression \(-276i^2\) becomes a positive real number, 276. This transformation helps us combine real terms separately from imaginary terms, which simplifies the entire expression to its final form. Understanding the behavior of \(i\) in these situations is essential, as it directly influences the result of operations involving complex numbers.