Problem 62 - Algebra 2

Question

Simplify. $$ (3+2 i)-(1-7 i) $$

Step-by-Step Solution

Answer

The simplified expression is \(2 + 9i\).

Step 1: Distribute the Negative Sign

The expression to simplify is \((3 + 2i) - (1 - 7i)\). First, distribute the negative sign across the second term: \(3 + 2i - 1 + 7i\).

Step 2: Combine Like Terms

Now, combine the real parts and the imaginary parts separately. The real parts are \(3 - 1\) and the imaginary parts are \(2i + 7i\).

Step 3: Simplify Real Parts

Subtract the real numbers: \(3 - 1 = 2\).

Step 4: Simplify Imaginary Parts

Add the imaginary numbers: \(2i + 7i = 9i\).

Step 5: Combine Results

Combine the results from Steps 3 and 4 to get the simplified expression: \(2 + 9i\).

Key Concepts

Imaginary Numbers

Imaginary numbers might sound a bit mysterious, but they're quite straightforward once you get the hang of them. They are built around the concept of \(i\), which is defined as the square root of -1. In mathematics, \(i^2 = -1\). This idea is crucial because it allows us to solve equations that don't have real number solutions. For instance, consider the equation \(x^2 = -1\). In the realm of real numbers, this equation has no solution because no real number squared equals a negative number. However, with imaginary numbers, \(x = i\) is a solution, since \(i^2 = -1\). Imaginary numbers are often used in engineering, physics, and complex mathematical problem-solving. They're an essential component of complex numbers, which combine real and imaginary parts.

Addition and Subtraction of Complex Numbers

Complex numbers are composed of two parts: a real part and an imaginary part. When adding or subtracting complex numbers, the process is quite intuitive if you handle the real and imaginary parts separately.To perform operations like addition or subtraction, follow these simple steps:

• Identify the real parts and add or subtract them.
• Do the same with the imaginary parts.
• Combine the results to form a new complex number.

Suppose we are simplifying the expression \((3+2i)-(1-7i)\). We can dissect it as follows:

• Real parts: \(3 - 1 = 2\)
• Imaginary parts: \(2i + 7i = 9i\)

Thus, the resulting complex number is \(2 + 9i\). This method allows complex number operations to be handled with ease, respecting real and imaginary separations.

Simplifying Complex Expressions

Simplifying complex expressions might appear daunting, but breaking it down into smaller steps can simplify the process. Remember, each component should be tackled systematically.Firstly, manage any subtraction or addition within parentheses by addressing any negative signs first, similar to distributing negative signs across a subtraction within parentheses. For example, in the expression \((3 + 2i) - (1 - 7i)\), distributing the negative gives:

3 + 2i - 1 + 7i

Next, group and simplify the expression:

• Combine \(3 - 1\) to get the real part.
• Combine \(2i + 7i\) for the imaginary part.

This results in

2 + 9i

Finally, always double-check your work by ensuring that all real and imaginary components are calculated and combined correctly. This ensures the expression is fully simplified, achieving a neat and concise final answer.