Problem 89 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \(4(5+3 i)\)

Step-by-Step Solution

Answer

The expression simplifies to \(20 + 12i\).

Step 1: Distribute the 4 through the parenthesis

Multiply each term inside the parentheses by 4. This gives:\[ 4 \cdot 5 + 4 \cdot 3i \]

Step 2: Perform Multiplications

Compute the multiplications from Step 1. This results in:\[ 20 + 12i \]

Step 3: Write in Standard Form

The expression is already in the form \(a + bi\), where \(a = 20\) and \(b = 12\). Thus, the simplified expression is:\[ 20 + 12i \]

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential math skill, especially when working with complex numbers. The goal is to make an expression easier to understand or work with by manipulating it to a simpler form. When dealing with complex numbers, expressions often involve real and imaginary components. Simplifying these expressions involves combining like terms and performing arithmetic operations to reduce the expression to its simplest form.

In our original exercise, we started with a complex expression, namely: \(4(5 + 3i)\).To simplify this expression, we distributed the 4 across both terms within the parentheses and performed the necessary multiplications. This process involves several steps and utilizes mathematical properties, such as the distributive property, to simplify the original complex expression.

Standard Form

Standard form for complex numbers is essential because it provides a universal way of expressing any complex number. This format makes it easier to understand and compare different complex numbers. Complex numbers are usually represented in the form \(a + bi\), where \(a\) is the real part, and \(b\)i represents the imaginary part.

In our example, after performing the calculations, we ended with \(20 + 12i\), which is clearly in the standard form. This makes it easy to identify the real component, which is \(20\), and the imaginary component, which is \(12i\). Writing in the standard form is not just a formality. It greatly aids in operations like addition, subtraction, or comparing complex numbers.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions easily. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then adding the results.

Mathematically, this property can be expressed as:\(a(b+c) = ab + ac\).In our example \(4(5 + 3i)\), the number \(4\) is distributed to both \(5\) and \(3i\), resulting in \(4  + 4 i = 20 + 12i\).This step simplifies the expression into two separate components, making it more manageable. Understanding the distributive property is crucial for simplifying complex numbers because it helps in breaking down expansions into smaller parts, making them easier to process.

Always remember, whenever you see parentheses with a multiplication outside, think about applying the distributive property to simplify your work!