Problem 7 - Algebra 2

Question

Simplify. $$ (6 i)(-2 i) $$

Step-by-Step Solution

Answer

The expression \((6i)(-2i)\) simplifies to \(12\).

Step 1: Identify the Expression

We need to simplify the expression \((6i)(-2i)\). Here, \(i\) is the imaginary unit, which satisfies \(i^2 = -1\).

Step 2: Multiply the Coefficients

First, multiply the real number coefficients: \(6\) and \(-2\). This gives \(6 \times -2 = -12\).

Step 3: Multiply the Imaginary Units

Now, multiply the imaginary units: \(i \times i = i^2\). Recall that \(i^2 = -1\).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3: \(-12 \times i^2 = -12 \times -1\). This simplifies to \(12\).

Step 5: Simplify the Expression

The simplified form of the given expression \((6i)(-2i)\) is \(12\).

Key Concepts

Imaginary Unit

In the realm of complex numbers, the imaginary unit, denoted as \(i\), is a fundamental concept. It is defined by the property that \(i^2 = -1\). This definition allows us to extend the real number system to include solutions to equations that would otherwise not be possible, such as finding the square root of a negative number.

• The imaginary unit \(i\) acts similarly to an algebraic variable when performing operations involving complex numbers.
• Understanding the properties of \(i\) is crucial for correctly manipulating expressions that involve complex numbers.

In our exercise, we encounter \(i\) while simplifying the expression \((6i)(-2i)\). Recognizing how \(i^2 = -1\) plays into the simplification process is key to arriving at the correct result.

Multiplying Coefficients

In expressions involving complex numbers, it's common to encounter a need to multiply coefficients. Coefficients are the real numbers paired with the imaginary unit, \(i\), in expressions like \(6i\) and \(-2i\).

• The first step in multiplying complex numbers is to focus on these real number coefficients. In the exercise provided, we multiply \(6\) and \(-2\) which equals \(-12\).
• This step simplifies the parts of the expression involving real numbers, separately from the imaginary unit.

Remembering to keep the multiplication of coefficients and the manipulation of the imaginary unit as separate tasks helps prevent errors and makes simplification cleaner and more efficient.

Simplification Process

The simplification process of expressions like \((6i)(-2i)\) involves several key steps that work together to resolve the expression into its simplest form.

• After multiplying the coefficients, the next step is to multiply the imaginary units together.
• When we multiply \(i \times i\), we use the property \(i^2 = -1\) to transition from a product of imaginary units to a real number.
• Combining results from separate calculations, namely the coefficient multiplication and the imaginary unit multiplication, will give us the final expression.

In this exercise, we get \(-12 \times i^2 = -12 \times -1\), which simplifies to \(12\). The process demonstrates not only the power of understanding complex multiplication but also the elegance of simplifying such expressions by carefully handling each part. By organizing and executing these steps, the originally complex expression is boiled down to a simple real number.