Problem 23 - College Algebra with Corequisite Support

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

-16 + 32i

Step 1: Distribute the Scalar to Each Term

The given expression is \[ (-2 + 4i) imes 8 \] We need to distribute the scalar 8 to each term inside the parentheses separately. This can be calculated as:\[ (-2) imes 8 + (4i) imes 8 \]

Step 2: Perform the Multiplication

Now, perform the multiplication for each term separately.1. Multiply \(-2\) by 8: \[ (-2) imes 8 = -16 \]2. Multiply \(4i\) by 8: \[ (4i) imes 8 = 32i \]

Step 3: Combine the Results

Combine the results from the multiplication:\[ -16 + 32i \]This is the result expressed as a simplified complex number.

Key Concepts

Complex Multiplication

Complex multiplication involves multiplying two complex numbers or a complex number and a real number together. Complex numbers take the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. The imaginary unit \( i \) is defined such that \( i^2 = -1 \). To multiply complex numbers, each term in the first number is multiplied by each term in the second number. If one of the numbers is real, it simplifies the process.

• Multiply the real parts together.
• Multiply the real number by the imaginary part.

When the second number is purely real (like 8 in our example), the multiplication becomes a simple distribution of the real number across the terms of the complex number. Hence, the expression \((-2+4i)(8)\) requires multiplying both \(-2\) and \(4i\) by 8 separately.

Scalar Distribution

Scalar distribution is a method used in mathematics to distribute a single number (scalar) across each term within parentheses. It's similar to the distributive property, allowing us to simplify expressions by multiplying the scalar with each part of an expression individually. With complex expressions, scalar distribution helps break down the problem into manageable pieces.

• Apply the scalar to each term in the complex number.
• Simplify separate expressions after multiplication.

For example, in the expression \((-2 + 4i) \times 8\), we multiply 8 by \(-2\) and 8 by \(4i\). This yields the products \(-16\) and \(32i\). This approach helps simplify expressions by treating each term separately before the final combination.

Simplification of Complex Expressions

Simplification of complex expressions involves combining and reducing the parts of a complex number to its most basic form. After performing operations such as multiplication and distribution, the expression might still contain unnecessary complexities that need simplifying.Once the separate calculations are completed, the next step is combining the results. In the example, after multiplying \(-2\times8 = -16\) and \(4i \times 8 = 32i\), the results are combined into a single complex number: \(-16 + 32i\).

• Ensure that like terms are combined.
• Double-check calculations to ensure simplification is accurate.

Finally, the expression is expressed neatly as one complex number in the form \( a + bi \), which in this case is \(-16 + 32i\). This orderly presentation makes further calculations or evaluations in complex problem-solving much clearer and easier to understand.