Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (5-2 i)(3 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(6 + 15i\).

Step 1: Identify the Distribution

To multiply the complex number \((5 - 2i)\) by the imaginary number \(3i\), we will use the Distributive Property, which means multiplying each term in the first pair by the term in the second.

Step 2: Distribute 3i into (5 - 2i)

Apply distribution by multiplying \(3i\) with each component of the complex number \((5 - 2i)\):\[3i imes 5 + 3i imes (-2i)\]

Step 3: Perform Each Multiplication

Calculate each part of the distribution:1. \(3i imes 5 = 15i\)2. \(3i imes (-2i) = -6i^2\)

Step 4: Substitute i² with -1

Recall that \(i^2 = -1\). Substitute in the expression to simplify:\(-6i^2 = -6(-1) = 6\).

Step 5: Combine Real and Imaginary Parts

Combine the real part and the imaginary part to get:\[6 + 15i\]

Step 6: Write the Final Answer

The simplified complex number is \(6 + 15i\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in mathematics used to simplify expressions and equations. It states that when you multiply a single term by a sum of terms, you must distribute the single term to each term within the parenthesis. For example, if you have an expression of the form \(a(b + c)\), you can simplify it by applying the distributive property as \(ab + ac\).

In the context of complex numbers, this property allows us to break down expressions of the form \((a - bi)(c)\), where you multiply each component of the first complex number by the second number. Particularly in our exercise, the distributive property helps in multiplying \(5 - 2i\) by \(3i\). Each part of the complex number—both real and imaginary—is multiplied by \(3i\). This step-by-step distribution eventually makes the calculation and simplification process much more manageable.

Imaginary Unit

In complex numbers, the imaginary unit \(i\) is a number that satisfies the equation \(i^2 = -1\). This definition is crucial because it allows us to extend our number system to include solutions to equations that do not have real number solutions.

For instance, in the multiplication \(3i \times (-2i)\) from our exercise, we encountered \(i^2\). By knowing that \(i^2 = -1\), we can replace \(-6i^2\) with \(6\). The imaginary unit thus transforms the expression significantly by flipping the sign, making handling complex numbers in calculations straightforward.

• It helps in acknowledging that the square root of negative numbers is possible in this extended number system.
• The power of \(i\) cycles every four powers, which is helpful in simplifying powers of \(i\).

Simplification of Complex Expressions

Simplification of complex expressions involves combining like terms and using fundamental properties such as \(i^2 = -1\). After distributing, you will often have terms that can be combined or simplified further.

In our exercise, after performing the distribution, we got two expressions \(15i\) and \(-6i^2\). Recognizing that \(-6i^2\) equals 6 allows us to combine this with the imaginary part \(15i\). Simplification, in this case, results in the expression \(6 + 15i\), where the first term represents the real part and the second term represents the imaginary part of the complex number.

Understanding how to efficiently rearrange and reduce expressions ensures that complex numbers are as easy to use and manipulate as real numbers. This skill involves:

• Evaluating powers of \(i\) correctly.
• Combining real parts and imaginary parts after simplification.
• Paying attention to the signs, since \(i\) can change them.