Problem 74 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(-5-3 i)(2-4 i)$$

Step-by-Step Solution

Answer

The product is \(2 + 14i\).

Step 1: Expand Using Distributive Property

To find the product, use the distributive property: \((-5 - 3i)(2 - 4i) = (-5)(2) + (-5)(-4i) + (-3i)(2) + (-3i)(-4i)\).

Step 2: Calculate Real Parts

Calculate the real part of the expression:\((-5)(2) = -10\) and \((-3i)(-4i) = 12\), then add these together (-10 + 12) = 2.

Step 3: Calculate Imaginary Parts

Calculate the imaginary part of the expression:\((-5)(-4i) = 20i\) and \((-3i)(2) = -6i\), then add these together: \(20i + (-6i) = 14i\).

Step 4: Combine to Standard Form

Combine the results from the real and imaginary parts calculated in steps 2 and 3 to express the answer in the standard form of a complex number:The standard form is \(a + bi\), which gives us \(2 + 14i\).

Key Concepts

Standard form

In complex numbers, the standard form is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, and \(bi\) represents the imaginary part, with \(i\) being the imaginary unit defined by the property \(i^2 = -1\). Writing complex numbers in this form makes it easier to perform operations like addition, subtraction, and multiplication, ensuring clarity in handling both real and imaginary components. For example, in the problem \((-5-3i)(2-4i)\), the final result in standard form is \(2 + 14i\). Keep in mind to arrange the number such that it clearly distinguishes between the real and imaginary parts.

Real and imaginary parts

Any complex number \(a + bi\) includes both a real part \(a\) and an imaginary part \(bi\). Understanding these components is crucial for working with complex numbers.

• Real Part (\(a\)): This is simply the numeric value without the imaginary unit \(i\). In our example from the exercise \(2 + 14i\), the real part is 2.


• Imaginary Part (\(bi\)): This includes the imaginary unit \(i\), where \(b\) is a real number. For the same example, the imaginary part is \(14i\).

Recognizing which part of the complex number is real and which is imaginary helps in performing operations and simplifying expressions, as demonstrated when combining terms after distributing in the product calculation.

Distributive property

The distributive property allows us to multiply a sum by multiplying each addend separately and then adding the products. It's essential for expanding products of complex numbers. When solving \((-5-3i)(2-4i)\), the distributive property is applied as follows:

• Multiply \((-5)\) by each term in \((2 - 4i)\) which gives \((-5)(2) + (-5)(-4i)\).


• Multiply \((-3i)\) by each term in \((2 - 4i)\) which results in \((-3i)(2) + (-3i)(-4i)\).

Combining all terms:

• \((-5)(2) = -10\)
• \((-5)(-4i) = 20i\)
• \((-3i)(2) = -6i\)
• \((-3i)(-4i) = 12\)

By organizing and combining the results, we derive the complex number in standard form \(2 + 14i\). This property not only simplifies the process but is fundamental in understanding complex multiplication.