Problem 80 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-3-6 i)^{2} $$

Step-by-Step Solution

Answer

The standard form is \(-27 + 36i\).

Step 1: Expand the Expression

The expression \((-3-6i)^{2}\) can be expanded as \((-3-6i)(-3-6i)\). This requires using the distributive property (also known as FOIL: First, Outer, Inner, Last).

Step 2: Apply the Distributive Property

Distribute each term in the first complex number to each term in the second. The calculation will be:First: \((-3)(-3) = 9\)Outer: \((-3)(-6i) = 18i\)Inner: \((-6i)(-3) = 18i\)Last: \((-6i)(-6i) = 36i^2\)

Step 3: Combine Like Terms

Combine the results from step 2:\(9 + 18i + 18i + 36i^2\).Remember that \(i^2 = -1\), so substitute it\(36i^2 = 36(-1) = -36\).

Step 4: Simplify to Standard Form

Now combine the real parts and the imaginary parts:The real part: \(9 - 36 = -27\).The imaginary part:\(18i + 18i = 36i\).Thus, the expression simplifies to: \(-27 + 36i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term with each term within a parenthesis. This property is particularly useful when dealing with expressions involving complex numbers, as it simplifies the multiplication process by breaking it down into smaller, easier-to-handle steps.

When you see an expression like \((-3-6i)^2\), you're actually looking at \((-3-6i)(-3-6i)\). The distributive property lets you multiply each term from the first complex number by each term in the second, ensuring no part of the expression is left out:

• Multiply the first terms: \((-3) imes (-3) = 9\)
• Multiply the outer terms: \((-3) imes (-6i) = 18i\)
• Multiply the inner terms: \((-6i) imes (-3) = 18i\)
• Multiply the last terms: \((-6i) imes (-6i) = 36i^2\)

This breakdown simplifies the task, making it easier to follow each calculation step-by-step.

FOIL Method

The FOIL method is a specific application of the distributive property used to multiply two binomials. "FOIL" stands for First, Outer, Inner, Last, which describes the order in which you multiply the terms in the binomials.

In our example, \((-3-6i)(-3-6i)\), each term is handled in its unique turn:

• First: The first terms in each binomial are multiplied: \((-3) imes (-3) = 9\)
• Outer: The outer terms from each binomial are multiplied: \((-3) imes (-6i) = 18i\)
• Inner: The inner terms are multiplied: \((-6i) imes (-3) = 18i\)
• Last: The last terms from each binomial pair are multiplied: \((-6i) imes (-6i) = 36i^2\)

Using the FOIL method provides a structured approach to ensure that all terms are accounted for, preventing oversight and making the process more systematic.

Standard Form

Complex numbers are often expressed in their standard form, which is written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined by the property that \(i^2 = -1\).

After applying the distributive property or FOIL method and performing the multiplications, the outcome is a collection of terms that needs to be combined appropriately. In our exercise, the expression simplifies to \(-27 + 36i\):

• First, you combine the real part terms: \(9 - 36 = -27\)
• Then, group the imaginary part terms: \(18i + 18i = 36i\)
• Thus, the standard form becomes: \(-27 + 36i\)

Standard form is essential as it allows easy identification of both the real and imaginary components of a complex number, facilitating further operations or solutions in more complex equations.