Problem 67 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(-6 i)(-2-7 i)$$

Step-by-Step Solution

Answer

The product is \(-42 + 12i\).

Step 1: Distribute the Terms

To solve this problem, we distribute the imaginary term \(-6i\) to each term inside the parenthesis. Essentially, you'll perform two multiplications here.- First, multiply \(-6i\) by \(-2\): \((-6i)(-2) = 12i\)- Second, multiply \(-6i\) by \(-7i\): \((-6i)(-7i) = 42i^2\)

Step 2: Simplify using the properties of i

Recall that the imaginary unit \(i\) is defined by \(i^2 = -1\). This property will help us simplify the expression further:- The term \(42i^2\) becomes: \[42 \times (-1) = -42\]So, the expression is: \[12i - 42\]

Step 3: Express in Standard Form

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.Our expression from Step 2 is:- Real part (\(a\)): \(-42\)- Imaginary part (\(b\)): \(12\)Thus, the product in standard form is: \[-42 + 12i\]

Key Concepts

Standard Form

The standard form of a complex number is expressed as \(a + bi\). Here, \(a\) is the real part, and \(b\) is the imaginary part. This form simplifies handling and understanding of complex numbers. In complex number expressions, separating the real and imaginary components makes it easier to read and solve.
For example, if you have a complex number like \-42 + 12i\, it's straightforward to say:

• Real part: -42
• Imaginary part: 12

This structure is especially useful when adding, subtracting, or comparing complex numbers.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). While \(i\) itself does not correspond to any real number, it allows us to extend the concept of numbers beyond the real line.
For example:

• \(i\)
• \(3i\)
• \(-5i\)

All these express imaginary numbers, where the coefficient of \(i\) is the imaginary part of a complex number. Using \(i\), you can multiply and manipulate expressions that would otherwise be impossible in the real number system.

Distributive Property

The distributive property is a key algebraic rule used when multiplying expressions to ensure every term is multiplied correctly. Essentially, you distribute one term across others in a bracket. For example, in the expression \((-6i)(-2-7i)\), distribute \(-6i\) to both \(-2\) and \(-7i\).
Using this property:

• \((-6i)(-2) = 12i\)
• \((-6i)(-7i) = 42i^2\)

This approach guarantees that every term is accounted for, leading to a complete and accurate result.

Multiplication of Complex Numbers

Multiplying complex numbers involves distributing each part of one number to both parts of another. For the expression \((-6i)(-2 - 7i)\), you firstly apply the distributive property: multiply \(-6i\) with each term in the brackets.
Perform operations like these:

• \(-6i \times -2 = 12i\)
• \(-6i \times -7i = 42i^2\)

Then, simplify using \(i^2 = -1\). The \(42i^2\) becomes \(-42\). The result is \(-42 + 12i\), which aligns neatly with the standard form \(a + bi\). By following these steps methodically, you can multiply any complex numbers successfully.