Problem 68 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(-9 i)(-4-5 i)$$

Step-by-Step Solution

Answer

-45 + 36i.

Step 1: Distribute the terms

Distribute (-9i) across each term in the complex number, (-4 - 5i). Start by multiplying (-9i) with (-4) to get 36i. Next, multiply (-9i) with (-5i) to get 45i^2.

Step 2: Simplify using the imaginary unit

Remember (i^2 = -1). So, replace 45i^2 with 45(-1), which simplifies to -45. The expression now becomes 36i - 45.

Step 3: Write in standard form

Combine real and imaginary components. The expression is now in standard form as (-45 + 36i).

Key Concepts

Standard Form

In complex numbers, expressing them in their standard form is crucial for clear communication and computation. The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here both \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. To express a complex number in standard form:

• Identify the real part, which is any term without the imaginary unit \(i\).
• Identify the imaginary part, which is the term that includes \(i\).
• Combine them into the form \(a + bi\).

Breaking down the complex number into real and imaginary components ensures that calculations involving complex numbers remain consistent and more accessible.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. By definition, \(i\) is the square root of \(-1\). This leads us to the important property that \(i^2 = -1\). Understanding \(i\) is essential when operating with complex numbers, especially during multiplication. Here’s how the imaginary unit works:

• When you multiply any term by \(i\), it transforms the term into an imaginary component.
• The square of \(i\), namely \(i^2\), is used to simplify expressions because \(i^2\) equals \(-1\).
• Knowing how to replace \(i^2\) with \(-1\) allows for simplifying complex expressions into their standard form.

This property is crucial when simplifying products of complex numbers, as seen in the solution exercise.

Distributive Property

The distributive property is a key principle in mathematics that also holds true for complex numbers. This property states that multiplying a single term across terms inside parentheses requires distributing the multiplicative factor to each term individually.For complex numbers, this process looks like:

• Apply the multiplicative term to each part of a complex number expression, typically given as \((a + bi)\).
• Calculate each multiplication separately, keeping track of real and imaginary components.
• Finally, combine them to create a simplified expression.

In the context of the exercise, using the distributive property involves multiplying \(-9i\) with each component of the complex number \(-4 - 5i\). This ensures that every part of the complex number is accounted for correctly in the final product.