Problem 21 - College Algebra

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 result is: \(6 + 15i\).

Step 1: Distribute the term

To solve the complex multiplication, first distribute the imaginary unit \(3i\) across each term in the binomial \((5 - 2i)\). This gives: \[5 \times 3i - 2i \times 3i.\]

Step 2: Simplify the Imaginary Multiplication

Next, simplify each multiplication separately. For the first term: \[5 \times 3i = 15i.\]For the second term, multiply the coefficients and remember that \(i^2 = -1\):\[-2i \times 3i = -6i^2 = -6(-1) = 6.\]

Step 3: Combine the Terms

Now, combine the results from Step 2 to express the product as a simplified complex number:\[6 + 15i.\]

Key Concepts

Complex Multiplication

Complex multiplication is the process of multiplying numbers where at least one number has an imaginary part. To multiply complex numbers, apply the distributive property, which involves multiplying each term in one complex number by each term in the other number. This can often feel like distributing terms in binomial equations.

For instance, in the original exercise, we had to multiply \(5 - 2i\) by \(3i\). Using the distributive property, each part of the first number was multiplied by \(3i\).

• Multiply the real part, \(5\), with the imaginary term, \(3i\).
• Then multiply the imaginary part, \(-2i\), with \(3i\).

These operations provide a systematic method to handle complex multiplication effectively.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\), which is something that doesn't exist within the set of real numbers. Therefore, \(i^2\), which represents the square of an imaginary unit, equals \(-1\).

This property is key when dealing with multiplication in complex numbers. When in the exercise, \(-2i\) was multiplied by \(3i\), we got \(-6i^2\). Applying the property that \(i^2 = -1\), this simplifies as follows: \(-6(-1) = 6\). It’s crucial to remember this identity because it often comes up, providing you the bridge to simplify complex products to a meaningful result.

Simplification of Complex Expressions

Simplification of complex expressions involves reducing complex numbers to their simplest form. This means combining like terms and applying any arithmetic rules that may help make the expression clearer.

Consider the solution of \(5 \times 3i - 2i \times 3i\) from the exercise. Once we multiplied each part, our expression looked like this: \(15i + 6\).

• Combine anything that looks alike, which often means combining terms with \(i\) as one part, and real numbers as the other.
• Since complex numbers have both real and imaginary components, ensure each is represented plainly in the final result.

By making sure terms are simplified and correctly combined, the final expression becomes clear and concise.

Complex Arithmetic

Complex arithmetic involves operations such as addition, subtraction, multiplication, and division with complex numbers. When performing these operations, it is crucial to deal with both the real and imaginary parts properly.

In general, arithmetic with complex numbers is straightforward, but remember to:

• Combine like terms, meaning work separately with the real parts and the imaginary parts.
• Handle multiplication with care, giving attention to distributing parts and simplifying using \(i^2 = -1\).

In solving complex arithmetic problems like the provided exercise, understanding each part's role ensures correct operation results. The goal is always to express the outcome efficiently and clearly, just as with regular real number arithmetic.