Problem 38 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ 5 i(4-7 i) $$

Step-by-Step Solution

Answer

The product is \(35 + 20i\).

Step 1: Distribute the Imaginary Number

To solve \(5i(4 - 7i)\), start by distributing the \(5i\) to both terms inside the parentheses. Multiply \(5i\) by \(4\) and by \(-7i\).

Step 2: Multiply the Real Number

First, multiply the real part: \(5i \times 4 = 20i\). This is straightforward since combining a real number with an imaginary number only results in an imaginary term.

Step 3: Multiply the Imaginary Number

Next, multiply the imaginary parts: \(5i \times (-7i) = -35i^2\). Since \(i^2 = -1\), this becomes \(-35(-1) = 35\).

Step 4: Combine the Results

Combine the results from the previous steps: the imaginary component \(20i\) and the real component \(35\).

Step 5: Write in Standard Form

Write the final answer in the standard form \(a + bi\) by placing the real component first: the result is \(35 + 20i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are an essential aspect of mathematics, especially when dealing with complex numbers. These are numbers that, when squared, yield a negative result. The imaginary unit is denoted by the symbol \(i\), and it is defined by the equation \(i^2 = -1\). This characteristic is what makes imaginary numbers unique and different from real numbers. In practical terms, whenever you encounter an expression like \(5i\), you are looking at an imaginary number where \(5\) is the coefficient and \(i\) represents the imaginary part. Operations with imaginary numbers follow similar rules as those with real numbers, keeping in mind that multiplying \(i\) by itself results in \(-1\).When working with imaginary numbers, it's crucial to understand:

• The basic definition: \(i = \sqrt{-1}\).
• Multiplication properties: for example, \( (a + bi)(c + di) = ac + bci + adi + bdi^2 \), where \(i^2 = -1\).
• The role of \(i^2\) in simplifying expressions: always convert \(i^2\) to \(-1\) to simplify expressions.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions and multiply terms within parentheses. It states that if you multiply a number by a sum, you need to distribute the multiplication to each term individually.For example, in the expression \(5i(4 - 7i)\), we apply the distributive property by multiplying \(5i\) with each term inside the parentheses.

• First, multiply \(5i\) with \(4\), which results in \(20i\).
• Then, multiply \(5i\) with \(-7i\), resulting in \(-35i^2\).

This property is particularly useful when dealing with polynomials or expressions with complex or imaginary numbers. In multiplication, these steps ensure that every element inside the parentheses is accurately considered. The resulting products can then be combined to form the simplified expression. Remembering the distributive property can simplify a wide range of mathematical problems, providing a powerful tool in both algebra and higher-level mathematics.

Standard Form of Complex Numbers

The standard form of complex numbers is crucial for properly expressing and understanding complex numbers. In this form, a complex number is typically written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.For instance, in solving \(5i(4 - 7i)\), after applying the distributive property, we find terms \(20i\) and \(-35i^2\). Recalling that \(i^2 = -1\), we simplify \(-35i^2\) to \(35\) so our expression becomes \(35 + 20i\). By expressing it in the standard form \(a + bi\), we can easily identify:

• The real component \(a = 35\).
• The imaginary component \(b = 20\), in the form \(20i\).

Writing complex numbers in this form allows for straightforward addition, subtraction, multiplication, and division of complex numbers.This format simplifies operations and comparison between complex numbers, making calculations more consistent and comprehensible.