Problem 54 - Intermediate Algebra

Question

$$ -5 i(-2+i) $$

Step-by-Step Solution

Answer

5 + 10i

Step 1: Distribute the Negative Sign

First, rewrite the expression by distributing the negative sign inside the bracket:\[-5 i (-2 + i) = -5i(-2) + (-5i)(i)\]

Step 2: Multiply Each Term

Multiply each term separately:- For the first term, multiply the constants and consider the sign: \[ -5i imes (-2) = 10i \]- For the second term, multiply the imaginary numbers: \[ -5i imes i = -5i^2 \]

Step 3: Simplify Imaginary Unit

Replace \(i^2\) with \(-1\), according to the property \(i^2 = -1\):\[-5i^2 = -5(-1) = 5\]

Step 4: Combine the Results

Combine the results from Step 2 and Step 3:\[10i + 5\]

Step 5: Final Answer

Rewrite the expression in standard form: \[5 + 10i\]

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the real number system. It satisfies the equation \(i^2 = -1\). This definition might seem unusual at first since there is no real number whose square is negative.

The introduction of \(i\) allows for the creation of complex numbers, which are numbers that have both a real and an imaginary part.

• Imaginary numbers are essential in many fields, especially in engineering and physics, for solving equations that do not have real solutions.
• Any real number multiplied by \(i\) becomes an imaginary number, for example, \(5i\) or \(-3i\).
• The imaginary unit \(i\) makes it possible to work with roots of negative numbers, such as \(\sqrt{-1} = i\).

Understanding the imaginary unit is crucial before diving into more complex calculations involving complex numbers.

Distributive Property

The distributive property is a fundamental principle in algebra that relates to how multiplication interacts with addition. It states that for any numbers \(a\), \(b\), and \(c\), \ a(b+c) = ab + ac.

This property is incredibly useful in simplifying mathematical expressions and solving equations. For example, in the exercise:

• The distributive property allows us to multiply each term inside the parentheses by \(-5i\).
• This means we consider each component separately: \(-5i \times (-2)\) and \(-5i \times i\).

Breaking down expressions using the distributive property can make complex equations much more manageable and less intimidating to work with.

Multiplication of Imaginary Numbers

Multiplying imaginary numbers involves using the imaginary unit \(i\), and sometimes the distributive property, depending on the complexity of the expression.

When multiplying \(i\) by another number, you treat it like a variable:

• As seen in the example, multiplying two imaginary numbers \(-5i \times i\) results in \(-5i^2\).
• Recall that \(i^2 = -1\), which transforms \(-5i^2\) into \(5\).

The important thing to remember is that when \(i^2\) appears, it should be replaced by \(-1\) immediately. This conversion is crucial for simplifying expressions correctly.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Writing complex numbers in this form makes them easier to understand and work with.

In our example:

• The final result \(5 + 10i\) is in the standard form, where \(5\) is the real part, and \(10i\) is the imaginary part.
• Standard form helps in identifying both components quickly, which is necessary for operations like addition, subtraction, and comparison.

Understanding and using the standard form allows for consistent procedures when performing mathematical operations on complex numbers, ensuring accuracy and clarity in communication.