Problem 23 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-2+4 i)(8) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-16 + 32i\).

Step 1: Distribute the multiplication

To multiply the complex number \((-2+4i)\) by 8, distribute 8 to both the real and imaginary parts of \((-2+4i)\). This gives: \[8 imes (-2) + 8 imes (4i)\].

Step 2: Multiply each part

Perform the multiplication. For the real part: \[8 imes (-2) = -16\]. For the imaginary part: \[8 imes 4i = 32i\].

Step 3: Combine the results

Combine the real and imaginary parts to form the complex number: \(-16 + 32i\). This is the simplified form of the complex number resulting from the multiplication.

Key Concepts

Multiplication of Complex Numbers

When multiplying complex numbers, it is essential to consider both the real and imaginary components of the numbers involved. A complex number typically takes the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In the exercise, we are tasked with multiplying a complex number (-2 + 4i) by a real number 8.
\[(-2 + 4i) \times 8\]
To perform this multiplication, you should distribute the real number across both the real and imaginary components of the complex number. This involves multiplying 8 with both -2 and 4i individually. This method ensures that each part of the complex number is being multiplied separately, maintaining their unique properties.

• The operation involves distributing, i.e., taking the real number and multiplying it by each part of the complex number.
• The process respects the structure of complex numbers, by operating separately on the real and imaginary components.

Using this approach guarantees that the integrity of the complex number is maintained throughout multiplication.

Distributive Property in Algebra

The distributive property is a fundamental concept in algebra that ensures each term inside a set of parentheses is multiplied separately by a number outside the parentheses. This property applies not only to simple numbers but also to algebraic expressions, including complex numbers.
In the given exercise, the distributive property helps to simplify the multiplication of the complex number (-2 + 4i) and the real number 8. It requires the separation of each component of the complex number, distributing the 8 across both \(-2\) and \(4i\).

• Using the distributive property, multiply the 8 with the first term: 8 \times (-2) = -16.
• Then, multiply the 8 with the imaginary term: 8 \times (4i) = 32i.

Thus, the operation is performed accurately and step-by-step, exemplifying the power of the distributive property in handling complex number operations.

Simplification of Complex Expressions

Simplifying complex expressions involves combining the real and imaginary parts after performing operations like multiplication. Once you've distributed and multiplied each part of the complex number by the real number, the next step is to combine these results into a single simplified expression.
In the exercise, after applying the distributive property and multiplying the terms, you obtain two separate results: -16 and 32i.
To combine them, connect the results as you would in a standard complex number, yielding \(-16 + 32i\).
This final step creates a simplified complex expression, preserving both components of the original complex number. It's crucial in mathematics to express the final solution in the standard form, \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

• Combining involves aligning both the real and imaginary parts together as a single expression.
• The simplified complex number reflects the end result of the operations performed, ready to be utilized in further mathematical contexts or applications.

Ensuring that these expressions are in the correct form is essential for clarity and accuracy in complex number calculations.