Problem 14 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(8-4 i)(-3+9 i)$$

Step-by-Step Solution

Answer

The product of the complex numbers \((8-4i)\) and \(-3+9i\) simplifies to \(12 + 84i\).

Step 1: Distribute

Distribute each part of the first complex number to the parts of the second: \[ (8-4i)(-3+9i) = 8*(-3) + 8*9i -4i*(-3) -4i*9i \]

Step 2: Simplify and gather like terms

Simplify and combine like terms: \[ = -24 + 72i +12i -36i^2 \]\n to give \[-24 + 84i -36i^2\]

Step 3: Convert the result using \(i^{2} = -1\)

Since \(i^{2} = -1\), the term -36\(i^{2}\) becomes +36: \[ -24 + 84i + 36 = 12 + 84i \].

Key Concepts

Distribute Property

The distribute property is a mathematical rule that helps us multiply numbers or expressions across groups. In the context of complex numbers, this property means that we need to multiply each part of one complex number by each part of another. For example, in the exercise (8-4i)(-3+9i), we applied the distribute property by taking each part of (8-4i) and multiplying it with each part of (-3+9i): - Multiply 8 by -3: 8*(-3) - Multiply 8 by 9i: 8*9i - Multiply -4i by -3: -4i*(-3) - Multiply -4i by 9i: -4i*9i Using the distribute property allows us to methodically break down and simplify expressions, ensuring that no part gets overlooked. By multiplying in order and combining like terms as shown in the steps, we not only use this property but also make calculations with complex numbers easier to manage.

Standard Form

Standard form for complex numbers is a way of expressing them as a sum of a real part and an imaginary part. The general format is: \[ a + bi \] where \(a\) is the real part and \(b\) is the coefficient of the imaginary unit \(i\). When asked to write a complex number in standard form, it is important to simplify the expression so that all like terms are combined, and the expression fits this pattern.In the solution to the exercise, after applying the distribute property and simplifying, we identify and separate the real terms from the imaginary terms:- The initial expression becomes -24 + 84i -36i^2- Since \(i^2\) equals -1, replace \(-36i^2\) with 36.This results in the complex number in standard form:\[ 12 + 84i \].This format is crucial as it simplifies calculations and communication of complex numbers across different mathematical applications.

Imaginary Unit

The imaginary unit, represented by \(i\), is a fundamental concept in complex numbers. It is defined as the square root of -1, making it different from any number on the real number line. Its key properties include:

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

These properties help in simplifying expressions involving powers of \(i\). In the exercise provided, using the property \(i^2 = -1\) is crucial when transforming terms from the distributed multiplication:Initially, we see the term \(-36i^2\). Since \(i^2 = -1\), \(-36i^2\) simplifies to +36 as shown: \[ -36 \cdot (-1) = 36 \]This simplification is an essential step when finding the standard form, Highlighting the unique properties of the imaginary unit helps us manage and interpret complex numbers accurately across various mathematical problems.