Problem 14 - College Algebra

Question

Find each product and write the result in standard form. $$ (8-4 i)(-3+9 i) $$

Step-by-Step Solution

Answer

The product of \((8-4i)\) and \((-3+9i)\) in standard form is \(12 + 84i\).

Step 1: Distribute the first set of terms

Multiply each term in the first complex number by each term in the second complex number: \((8 * -3) + (8 * 9i) + (-4i * -3) + (-4i * 9i)\)

Step 2: Perform the multiplication

Perform the multiplication for each term: \(-24 + 72i + 12i - 36i^2\)

Step 3: Simplify

As \(i^2\) is equals to -1, replace \(i^2\) with -1 and combine like terms: \(-24 + 72i + 12i + 36 = 12 + 84i\) which is the result in standard form.

Key Concepts

Multiplication of Complex Numbers

Multiplying complex numbers may seem like a complex task - pun intended - but it's quite like multiplying binomials. The key here is to use the distributive property, which allows you to break down the multiplication into manageable steps. This approach is often referred to as the "FOIL method" when multiplying two binomials, where each letter stands for First, Outside, Inside, and Last, representing the pairs of terms you'll multiply.

• First: Multiply the first terms from each complex number.
• Outside: Multiply the outer terms from each complex number.
• Inside: Multiply the inside terms.
• Last: Multiply the last terms.

The multiplication \[(8-4i)(-3+9i)\]begins by distributing each term from the left complex number into the right complex number. The calculation structure becomes:

• First: \(8 \times -3 = -24\)
• Outside: \(8 \times 9i = 72i\)
• Inside: \(-4i \times -3 = 12i\)
• Last: \(-4i \times 9i = -36i^2\)

This step-wise approach simplifies the complex multiplication into smaller, more manageable parts.

Standard Form of Complex Numbers

A complex number is usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) represents the real part, while \(bi\) contains the imaginary part, with \(i\) being the imaginary unit \(i = \sqrt{-1}\). Writing complex numbers in this standard form makes them easier to read and manipulate in mathematical operations.
In our example, after performing the multiplication, we obtained: \[-24 + 72i + 12i - 36i^2\]The term \(-36i^2\) simplifies because \(i^2 = -1\), so it becomes \(36\).After combining like terms, which are the real numbers and the imaginary coefficients, you should get \(12 + 84i\), which is the standard form of the result. This form allows you to understand clearly the real and imaginary contributions to the number, making it practical for further calculations.

Distributive Property

The distributive property is a fundamental concept in algebra that makes multiplication over addition more straightforward. It states simply that:For any real numbers \(a, b, c\): \[a(b + c) = ab + ac\]This property is vital in simplifying expressions and is extensively used in multiplying complex numbers.

• In the expression \/ step -- \((8-4i)(-3+9i)\)
• We distribute each term in the first complex number to each term in the second complex number.
• This breaks the problem down into smaller sums that are easier to handle.

The distributive property, thus, allows you to focus on individual multiplications, and then add all the components together to get the final product. This technique doesn't just work with numbers, but is also applicable in various algebraic structures, making it a tool to simplify multiplication across mathematics.