Problem 40 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -5 i(-2+i) $$

Step-by-Step Solution

Answer

The answer is \(5 + 10i\).

Step 1: Distribute the Terms

Apply the distributive property of multiplication to the expression \(-5i(-2+i)\). This means multiplying \(-5i\) by each term inside the parentheses. Essentially, you will perform the following multiplications: \(-5i\times(-2)\) and \(-5i\times i\).

Step 2: Multiply Each Term

First, multiply \(-5i\) by \(-2\):\[(-5i) imes (-2) = 10i\]Next, multiply \(-5i\) by \(i\):\[(-5i) imes i = -5i^2\]

Step 3: Simplify Using \(i^2 = -1\)

Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\) in the expression:\[-5i^2 = -5(-1) = 5\]

Step 4: Combine Like Terms

Combine the results from the previous steps. The expression becomes:\[5 + 10i\]This is in the form \(a + bi\), where \(a = 5\) and \(b = 10\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in arithmetic and algebra, which allows us to multiply a single term across a set of terms inside parentheses. When faced with an expression like \(-5i(-2+i)\), we use the distributive property by distributing the multiplication of \(-5i\) to both terms inside the parentheses: \(-2\) and \(i\).

This process involves breaking down the multiplication into simpler, manageable parts:

• Multiply \(-5i\) by \(-2\), resulting in \(10i\).
• Then, multiply \(-5i\) by \(i\), leading to \(-5i^2\).

By distributing the \(-5i\), we turn a complex multiplication problem into a simpler series of multiplications. This step is crucial for organizing your work and ensuring an accurate solution.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the notion of real numbers into the complex plane. It is defined by the property \(i^2 = -1\). This unique characteristic is what sets imaginary units apart from real numbers.

In the context of multiplying complex numbers, recognizing the role of \(i\) is essential. For example, when multiplying \(-5i\) by \(i\), it results in \(-5i^2\). Utilizing the property \(i^2 = -1\), we convert \(-5i^2\) to \(5\).

This substitution is a critical step in simplifying expressions involving complex numbers and their imaginary components. Understanding how \(i\) operates allows you to work effectively with complex numbers and fully grasp their unique characteristics.

Multiplication of Complex Numbers

Multiplying complex numbers involves using several rules from basic algebra, especially focusing on combining terms and dealing with imaginary units. Let's see how this works with the expression \(-5i(-2+i)\).

When you multiply these complex components:

• First, the multiplication of \(-5i\) with the real component \(-2\) gives \(10i\).
• The multiplication with the imaginary component \(i\) gives \(-5i^2\).

The key is the special property of the imaginary unit, where \(i^2\) converts to \(-1\). This simplifies \(-5i^2\) to \(5\), allowing you to combine and reorder terms to achieve the expression \(5 + 10i\).

This result follows the standard complex number form \(a + bi\). Mastering the art of multiplying complex numbers requires clear comprehension of both the distributive law and the implications of \(i^2 = -1\). It’s about recognizing how real and imaginary parts interact and transform, leading to your final answer.