Problem 106 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ -9 i(4-6 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-54 - 36i\).

Step 1: Distribute the Complex Number

To start, distribute \(-9i\) across the terms inside the parentheses.\(-9i(4-6i) = (-9i)(4) + (-9i)(-6i)\).

Step 2: Multiply Each Term

Calculate each multiplication separately:- For \((-9i)(4)\), we have \(-36i\).- For \((-9i)(-6i)\), we notice this is a product of two imaginary numbers. The result is \(54i^2\).

Step 3: Simplify Using the Property of \(i^2\)

Remember that \(i^2 = -1\). Therefore, \(54i^2 = 54(-1) = -54.\)

Step 4: Combine Real and Imaginary Parts

Combine the real part (-54) and the imaginary part (-36i), giving us the expression \(-54 - 36i\).

Key Concepts

Understanding the Distributive Property in Complex Numbers

The distributive property is a fundamental aspect of mathematics that helps you expand expressions. With complex numbers, it works just like with real numbers. When you have an expression like \(-9i(4 - 6i)\), you apply the distributive property by multiplying \(-9i\) with each term inside the parentheses separately.

Applying this property, you perform the operation:

• Multiply \(-9i\) by 4.
• Then multiply \(-9i\) by \(-6i\).

This results in two separate terms that are further simplified in the solution process.
The distributive property simplifies the handling of expressions involving both real and imaginary terms, making complex arithmetic more structured.

The Imaginary Unit: i

The imaginary unit, often denoted as \(i\), is the backbone of complex numbers. It allows us to extend the real number system to solve equations that would otherwise have no real solution. By definition, \(i\) is the square root of \(-1\). Therefore, \(i^2 = -1\), which is a crucial identity used in calculations.

When dealing with expressions like \(-9i(-6i)\), you multiply the two imaginary units together. This operation uses the property \(i^2 = -1\), transforming the expression into a real number. Understanding \(i\) and this property helps simplify expressions and solve complex equations effectively.

Real and Imaginary Parts of Complex Numbers

In a complex number, the real part is completely separate from the imaginary part. A complex number takes the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. For example, in the expression\(-54 - 36i\),

• -54 is the real part.
• -36i is the imaginary part.

When you're solving problems involving complex numbers, it’s important to identify and keep the real and imaginary parts distinct until the final step. This distinction helps in cleaning up expressions and combining like terms later on.

Simplification of Complex Expressions

Simplification is crucial when working with complex numbers. When you carry out operations on complex expressions, like in the given exercise, you often end up with a mixture of real numbers and terms involving the imaginary unit \(i\).

In this process:

• Combine all real parts together.
• Combine all imaginary parts and express them as a single term.

For instance, after distributing and multiplying, you simplify using \(i^2 = -1\) to switch some imaginary products to real numbers. Finally, you combine all real numbers into one part and all terms with \(i\) into the imaginary part of the solution. In our example, \(-54 - 36i\) is the simplest form that neatly organizes both components.