Problem 30 - Algebra 2

Question

Simplify. $$ (-2 i)(-6 i)(4 i) $$

Step-by-Step Solution

Answer

-48i

Step 1: Simplify the Product of Two Terms

First, simplify the product of the first two terms: \[(-2i) \cdot (-6i)\]. Use the rule \((a \cdot b) \cdot i^2 = a \cdot b \cdot (-1)\) since \(i^2 = -1\). Thus, this simplifies to: \[12i^2 = 12(-1) = -12.\] The result is \(-12\).

Step 2: Multiply by the Remaining Term

Now multiply the result from Step 1 by the last term: \[(-12) \cdot (4i).\] This results in \(-48i\) since \(-12 \times 4 = -48\), and \(i\) is carried along.

Step 3: Conclusion: Simplified Expression

Thus, the expression \((-2i)(-6i)(4i)\) simplifies to \(-48i\).

Key Concepts

Imaginary Unit

In the world of mathematics, the imaginary unit is a critical concept when dealing with complex numbers. The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\). This is a unique characteristic because, in the realm of real numbers, no number squared results in a negative. Thus, \(i\) serves as the foundation for constructing complex numbers.
Complex numbers extend the real numbers by incorporating this unit \(i\). A typical complex number is written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, and \(bi\) is the imaginary part.
When you multiply \(i\) by itself, you use its defining property to find that \(i^2 = -1\). This is especially important when simplifying expressions involving complex numbers.

Multiplication of Complex Numbers

The multiplication of complex numbers can involve more than just applying the distributive property. It can also involve the imaginary unit \(i\) and its properties. Let's say we need to multiply multiple complex terms like \((-2i)(-6i)(4i)\).
Start by multiplying the first two terms: \((-2i) \times (-6i)\). To do this, you treat \(i\) just like a variable but remember its key property: \(i^2 = -1\). So the multiplication goes as:

• First, multiply the coefficients: \((-2) imes (-6) = 12\)
• Then multiply \(i \times i = i^2\).

Based on the property \(i^2 = -1\), replace \(i^2\) with \(-1\), resulting in \(12 \times -1 = -12\).
This approach allows you to handle multiplication when complex numbers with \(i\) are involved continually.

Simplification Process

The simplification process of complex expressions involves reducing products involving imaginary numbers to their simplest form.
Once you've multiplied the first two complex numbers, you often need to incorporate additional terms. Take the product result \(-12\) and the remaining complex term \(4i\).
Proceed by handling the multiplication \((-12) \times (4i)\):

• Multiply the real number by the imaginary part: \(-12 \times 4 = -48\).
• Since \(i\) remains as a factor, the result becomes \(-48i\).

This type of simplification is common when dealing with multiple imaginary components. It breaks down what may initially seem a complex operation into manageable steps.