Problem 53 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ -3 i(-1+9 i) $$

Step-by-Step Solution

Answer

The result is \(27 + 3i\).

Step 1: Distribute the Terms

Start by applying the distributive property to expand the expression \(-3i(-1 + 9i)\).This means you multiply \(-3i\) by each term inside the parentheses:\(-3i imes -1 + (-3i) imes (9i)\).

Step 2: Multiply the Terms

Calculate each product from the expansion in step 1:1. \(-3i imes -1 = 3i\).2. \((-3i) imes (9i) = -27i^2\).This gives us the expression:\(3i - 27i^2\).

Step 3: Simplify Using \(i^2 = -1\)

Remember that \(i^2 = -1\).So, \(-27i^2 = -27(-1) = 27\).Now the expression becomes:\(27 + 3i\).

Step 4: Write in the Form \(a + bi\)

The expression \(27 + 3i\) is already in the form \(a + bi\), where \(a = 27\) and \(b = 3\).

Key Concepts

Distributive Property

The distributive property is a fundamental rule of algebra that allows us to simplify expressions. When we apply the distributive property, we multiply each term inside the parentheses by the term outside. This property is particularly useful with expressions that contain variables or complex numbers.
In the context of complex numbers, the distributive property works the same way as it does with real numbers. For example, in the expression \(-3i(-1 + 9i)\), we distributed \(-3i\) to each term inside the parentheses.

• First, multiply \(-3i\) by \(-1\), resulting in \(-3i \times -1 = 3i\).
• Next, multiply \(-3i\) by \(9i\), resulting in \((-3i) \times (9i) = -27i^2\).

By using the distributive property, we expanded a single expression into two separate terms, making it easier to compute.

Imaginary Unit

The imaginary unit \(i\) is a unique concept in mathematics. It is the cornerstone of complex numbers, providing a way to extend the concept of square roots to negative numbers. The imaginary unit is defined as \(i = \sqrt{-1}\), which means that \(i^2 = -1\).
Understanding how the imaginary unit operates with other numbers is vital. In our exercise, we encounter \(-27i^2\), a term that involves \(i^2\). By substituting for \(i^2\), we replace it with \(-1\). Therefore: \(-27i^2 = -27 \times (-1) = 27\).

• Remembering that \(i^2 = -1\) is critical since it allows us to transform and simplify complex number expressions.
• This ensures our final result is in the standard form of a complex number.

Complex Multiplication

Complex multiplication involves multiplying numbers that have both real and imaginary components. For two complex numbers \(a + bi\) and \(c + di\), the multiplication process is:

• Expand the expression: \(ac + adi + bci + bdi^2\).


• Simplify using \(i^2 = -1\): The expression becomes \(ac + (ad + bc)i - bd\).
• Combine like terms, leading to the result in the form \(A + Bi\).

In our problem, we perform complex multiplication using the single imaginary term \(-3i\). To multiply, we treated each part of the expression within the parentheses separately, giving us:

• Multiply \(-3i\) by a real number \(-1\) to gain \(3i\).
• Multiply \(-3i\) by the imaginary number \(9i\), resulting in \(-27i^2\).

Remember, working with complex numbers requires careful attention to both real and imaginary parts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. With complex numbers, these expressions become richer, as they incorporate the imaginary unit \(i\) alongside real numbers. The knowledge of how to work with algebraic expressions prepares us for handling more intricate mathematics.
In our exercise, the goal was to simplify the expression using known algebraic procedures. The process followed:

• Apply the distributive property to each term.


• Simplify the expression using \(i^2 = -1\).
• Combine terms to arrive at the form \(a + bi\).

Beyond the exercise, recognizing the structure and manipulation of algebraic expressions involving complex numbers helps solve real-life problems in engineering, physics, and beyond. The ability to manage both numerical and imaginary components offers a powerful toolkit for tackling a wide range of applications.