Problem 70 - Algebra 2

Question

Simplify. \((7+2 i)(1-i)\)

Step-by-Step Solution

Answer

The simplified expression is \(9 - 5i\).

Step 1: Distribute the Terms

To simplify the expression \((7 + 2i)(1 - i)\), we need to apply the distributive property, which involves multiplying each term in the first bracket by each term in the second bracket. Start by expanding as follows: \[(7)(1) + (7)(-i) + (2i)(1) + (2i)(-i)\].

Step 2: Perform Basic Multiplication

Carry out each multiplication from the expanded form:1. \(7 \times 1 = 7\)2. \(7 \times (-i) = -7i\)3. \(2i \times 1 = 2i\)4. \(2i \times (-i) = -2i^2\)

Step 3: Simplify Using "i" Properties

Recall that \(i^2 = -1\). Substitute \(-2i^2\) with \(-2(-1) = 2\). The expression becomes:\[7 - 7i + 2i + 2\].

Step 4: Combine Like Terms

Combine the real parts and the imaginary parts:- Real parts: \(7 + 2 = 9\)- Imaginary parts: \(-7i + 2i = -5i\)Thus, the simplified expression is: \[9 - 5i\].

Key Concepts

Distributive Property

The Distributive Property is an essential foundational concept when dealing with expressions, especially in algebra. It allows you to multiply a single term by each term in a parentheses. In this exercise, we have an expression

• do the distribution by breaking it down into smaller problems.
• the expression \((7+2i)(1-i)\) expands using \(7\) and \(2i\) via the distributive property to \[(7)(1) + (7)(-i) + (2i)(1) + (2i)(-i)\].

Effectively, each term in the first bracket is distributed across every term in the second bracket. By mastering the distributive property, you make it simpler to tackle complex multiplication problems in algebra. The rule here allows for addition and multiplication to be correctly simplified and expanded efficiently.

Imaginary Unit

The Imaginary Unit, denoted by \(i\), is a core element in complex numbers. It is defined by the property that \(i^2 = -1\). This particular characteristic plays a significant role when handling complex numbers and operations that involve them.
In our simplification task, understanding the imaginary unit is critical to properly handling the term \(-2i^2\). Remember:

• having \(i^2 = -1\), transforms the expression \(-2i^2\) to \( 2 \).

This concept is pivotal when simplifying expressions, as it helps convert complicated terms involving \(i\) into real numbers by using the specific property of \(i\). Mastering this ensures you can blend real and imaginary numbers seamlessly in your calculations.

Simplifying Expressions

Simplifying Expressions is a crucial skill in mathematics that involves rewriting an expression in the most concise or manageable form possible. We see this process in action when we simplify \((7+2i)(1-i)\) to reach the answer \(9 - 5i\). One key stage in working through these expressions is to:

• First, expand the expression using methods like the distributive property.
• Carry out multiplications carefully while keeping in mind the properties of the imaginary unit.
• Combine like terms, including real numbers and imaginary numbers (terms with \(i\)).

The goal of simplification is a streamlined result, and it didn't happen until:

• focusing on combining terms like \(7 + 2 = 9\)
• grouping \(-7i + 2i = -5i\).

Understanding this allows tackling similar problems with greater efficiency and confidence.

Real and Imaginary Parts

In complex numbers, every number contains two parts: the real and the imaginary part. These components are fundamental when simplifying expressions or performing operations involving complex numbers.
For instance, in the expression \(9 - 5i\), the \(9\) is the real part, and \(-5i\) is the imaginary part.

• Adding real parts together;
• combining imaginary parts helps ensure clarity when finalizing expressions.

In the simplification process, the different nature of real and imaginary parts requires careful handling:

• Real numbers are the standard numbers without \(i\), representing an actual quantity.
• Imaginary numbers are those that include \(i\), adjusting a numerical expression with altered mathematics methodologies.

Grasping the difference between these two parts aids in clearly organizing numbers and leads to an accurate simplified expression.