Problem 22 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (5-3 i)(1+i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(8 + 2i\).

Step 1: Distribute Terms

Start by applying the distributive property to the expression \((5-3i)(1+i)\). This means we need to multiply each term in the first binomial by each term in the second binomial. This gives us:\[5(1) + 5(i) - 3i(1) - 3i(i)\]

Step 2: Simplify the Real Parts

Next, let's simplify the real parts of the expression obtained from multiplication. The real parts are \(5(1)\) and \(-3i(i)\) which simplifies as follows:\[5 \text{ and } -3i^2\]Since \(i^2 = -1\), we have:\[-3(-1) = 3\]Thus, add the numbers: \[5 + 3 = 8\]

Step 3: Simplify the Imaginary Parts

Now, simplify the imaginary parts of the expression. The imaginary parts are:\[5(i) - 3i(1) = 5i - 3i\]Calculate: \[5i - 3i = 2i\]

Step 4: Combine Real and Imaginary Parts

Combine the real and imaginary parts simplified from the above steps to form the final result. From Steps 2 and 3, the expression simplifies to:\[8 + 2i\]Thus, the expression \((5-3i)(1+i)\) evaluates to \(8 + 2i\).

Key Concepts

Distributive Property

In the world of mathematics, the distributive property is a fundamental principle that allows us to simplify expressions and perform calculations with ease. The distributive property states that multiplying a sum by a single term is equivalent to multiplying each addend by the term and then summing the products. To put it simply,

• The principle is used to expand expressions like \(a(b + c) = ab + ac\)
• This same concept helps in dealing with binomials, such as \((5-3i)(1+i)\)

By distributing each term of the first binomial \((5-3i)\) over each term of the second binomial \((1+i)\), we break the problem into smaller, more manageable parts. Therefore, applying this property is crucial for accurately expanding and simplifying mathematical expressions.

Imaginary Unit

In mathematics, the imaginary unit is usually denoted as \(i\), which stands for the square root of -1. This is a key component in complex numbers, where we express numbers in the form \(a + bi\). Here,

• \(a\) represents the real part
• \(b\) represents the coefficient of the imaginary unit

One important property to remember about the imaginary unit is that \(i^2 = -1\). This understanding allows us to manipulate expressions involving imaginary numbers by converting products of \(i\) into real numbers where possible. In our example, knowing that \(-3i(i)\) becomes \(-3i^2\), we can replace \(i^2\) with -1 and simplify effectively. This insight into the nature of \(i\) is crucial when working through complex numbers and provides the necessary foundation for solving these expressions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form by performing operations and combining like terms. This process involves:

• Identifying and grouping real terms
• Identifying and grouping imaginary terms

In our example, after distributing terms using the distributive property, we are left with several components: real terms like \(5\) and imaginary terms such as \(-3i\). To simplify, it's crucial to perform the operations separately for real parts and imaginary parts.

• First, focus on the real parts: Combine \(5\) and \(3\) (obtained from \(-3i^2\) where \(i^2 = -1\)), resulting in \(8\).
• Next, handle the imaginary parts: Combine \(5i\) and \(-3i\) to get \(2i\).

The final step in this simplification process is recombining the simplified real part and imaginary part, resulting in a final expression in the form \(a + bi\). In this case, our expression simplifies neatly to \(8 + 2i\). Understanding how to reduce complex expressions to their simplest form is a skill that makes working with algebra much more manageable.