Problem 105 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ 6 i(2-3 i) $$

Step-by-Step Solution

Answer

The final result is \( 18 + 12i \).

Step 1: Distribute the Imaginary Unit

Expand the expression by distributing the imaginary unit across the terms within the parenthesis:\[ 6i(2 - 3i) = 6i \cdot 2 + 6i \cdot (-3i) \] This results in two separate multiplications: \(6i \cdot 2\) and \(6i \cdot (-3i)\).

Step 2: Perform the Distribution Multiplications

Calculate each part of the distribution separately:For the first term: \[ 6i \cdot 2 = 12i \]For the second term:\[ 6i \cdot (-3i) = -18i^2 \]Since \(i^2 = -1\), this becomes:\[ -18(-1) = 18 \]

Step 3: Combine the Real and Imaginary Parts

Now combine the results from Step 2. We have a real part and an imaginary part:Real part: \(18\)Imaginary part: \(12i\)Finally, combine them to reach the final answer:\[ 18 + 12i \]

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), stands as a fundamental building block in the world of complex numbers. It is defined by the property \(i^2 = -1\). This definition is crucial because it allows us to perform operations that involve the square root of negative numbers.

Here’s a breakdown of the imaginary unit's magic:

• Understanding \(i\): Its main role is transforming the square root of negative numbers into a form we can work with in mathematics.
• Simplifying Expressions: The imaginary unit enables the simplification of expressions involving roots of negative numbers, turning mathematical hurdles into tangible solutions.
• Real-world Applications: It's used in diverse fields, including engineering and physics, to solve real-world problems that require complex number solutions.

By using \(i\), complex numbers can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary component.

Distribution Property

The distribution property is fundamental in mathematics and ensures that each term within a bracket is multiplied individually by the outside term. This property is often expressed as \( a(b + c) = ab + ac \).

In the realm of complex numbers and expressions involving the imaginary unit, like our example: 6i(2 - 3i), the distribution property takes on an essential role:

• Step-by-step Multiplication: The term outside the parentheses (in this case, 6i) multiplies each term inside separately. This yields two separate expressions: \(6i \times 2\) and \(6i \times (-3i)\).
• Breaking Complexity: By addressing each part of the expression separately, we simplify the calculation process. This leads to a manageable pathway to reach an overall solution.

The distribution property is powerful and versatile, making complex calculations more systematic and approachable for students and professionals alike.

Multiplication of Complex Numbers

Multiplying complex numbers involves both real and imaginary components, which can often seem daunting. However, by systematically applying known mathematical properties, this process can be simplified and understood easily.

Consider the operations with complex numbers as in our previous exercise:

• Standard Operation: When multiplying two complex numbers, each part of the first number multiplies with each part of the second number.
• Special Case with \(i\): Notably, when \(i\) multiplies itself (\(i \cdot i\)), it results in \(-1\) due to the identity \(i^2 = -1\).
• Real and Imaginary Components: The final outcome must be distinguished into real and imaginary parts, as seen in combining \(18\) (real) and \(12i\) (imaginary) to reach \(18 + 12i\).

Remember, each multiplication step with complex numbers is guided by these principles, ensuring that the expression remains balanced and accurate. The more you practice, the more intuitive multiplying complex numbers will become.