Problem 109 - Algebra 2

Question

Simplify each expression. $$ (2-3 \sqrt{-4})+(4+2 \sqrt{-16}) $$

Step-by-Step Solution

Answer

The simplified expression is \(6 + 2i\).

Step 1: Simplify the square roots

Starting with the given expression \( (2 - 3 \sqrt{-4}) + (4 + 2 \sqrt{-16}) \), we simplify the square roots. Since the square root of \( -x \) is \( \sqrt{x} \times i \), we can simplify \( \sqrt{-4} \) as \( \sqrt{4} \times i = 2i \). Similarly, \( \sqrt{-16} \) can be simplified to \( \sqrt{16} \times i = 4i \). Then inserting these simplified square roots into the expression, we have \( (2 - 3 \times 2i ) + (4 + 2 \times 4i) \).

Step 2: Perform the Multiplication

Our next step is to perform the multiplication operations. So our expression becomes \( (2 - 6i) + (4 + 8i) \).

Step 3: Perform the Addition

Finally, perform the addition operations to combine the real and imaginary parts separately. So, the simplified expression is \( (2 + 4) + ( - 6i + 8i ) = 6 + 2i \).

Key Concepts

Imaginary Numbers

Imaginary numbers revolve around the central concept of the imaginary unit, denoted by the symbol \(i\), which is defined as the square root of -1. Imaginary numbers arise when dealing with negative numbers inside square roots. For instance, when we encounter \(\sqrt{-4}\), we realize that it cannot be simplified within the set of real numbers. Hence, we employ the imaginary unit: \(\sqrt{4} \times i = 2i\).

Imaginary numbers are used in many areas of mathematics and engineering to extend real numbers, especially in solving equations that have no real solutions. They help us to represent numbers that are not present on the traditional number line but exist in a plane called the "complex plane."

• When handling negative square roots, remember: \(\sqrt{-x} = \sqrt{x} \times i\).
• Use \(i\) to simplify expressions involving negative square roots.
• The notation \(i\) is fundamental in forming complex numbers.

Make sure to become comfortable with imaginary numbers, as they play a vital role in simplifying expressions like the one in the problem discussed.

Simplifying Expressions

Simplifying expressions involves breaking down mathematical expressions into their simplest form. In the context of complex numbers and imaginary numbers, this often means expressing numbers in a format that combines both real and imaginary components. For example, take the expression \((2 - 3 \sqrt{-4}) + (4 + 2 \sqrt{-16})\). To simplify it, we first simplify any square roots that involve negative numbers, converting them using the \(i\) unit.

After simplifying the square roots, we replace them in the original expression to get real and imaginary terms. The given expression became \((2 - 6i) + (4 + 8i)\) once the square roots were simplified.

In general, simplifying expressions follows these steps:

• Identify and simplify any complex components, such as square roots of negative numbers.
• Combine like terms, which means adding or subtracting the real parts separately from the imaginary parts.
• Write the result as a single simplified expression that reflects all operations performed.

These steps help streamline mathematical expressions, making them easier to interpret and use in further calculations.

Algebraic Operations

Algebraic operations refer to the basic mathematical processes of addition, subtraction, multiplication, and division applied to numbers, including complex numbers. When it comes to handling complex numbers, it is important to perform these operations by isolating real and imaginary parts. In our exercise, we need to add two complex expressions, \((2 - 6i) + (4 + 8i)\).

To add these complex expressions:

• Add the real components together: \(2 + 4 = 6\).
• Add the imaginary components together: \(-6i + 8i = 2i\).

This results in a final simplified expression of \(6 + 2i\).

When performing algebraic operations on complex numbers, focus on:

• Separating operations for real and imaginary parts.
• Being careful with signs, especially when subtracting and dealing with negative numbers.
• Practicing with different sets of numbers to gain fluency.

Understanding how to perform these operations correctly is crucial, as it serves as a foundation for more advanced work in algebra and analysis.