Problem 56 - Elementary and Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i\) See Example 5 . $$ 2 i(7+2 i) $$

Step-by-Step Solution

Answer

The result is \(-4 + 14i\).

Step 1: Distribute 2i

To multiply the expression \(2i(7+2i)\), we need to distribute \(2i\) to each term within the parentheses. This gives us two separate multiplications to evaluate: \(2i \cdot 7\) and \(2i \cdot 2i\).

Step 2: Calculate each multiplication

Let's start with \(2i \cdot 7\). Multiply the real number by the imaginary unit: \(2i \cdot 7 = 14i\).Next, calculate \(2i \cdot 2i\). Remember that \(i^2 = -1\), so: \[2i \cdot 2i = 4i^2 = 4(-1) = -4\].

Step 3: Combine the results

Now we combine the results from Step 2: \[14i + (-4)\]. This simplifies to \(-4 + 14i\).

Step 4: Express in form \(a+bi\)

Write the complex number \(-4 + 14i\) in the form \(a+bi\), where \(a=-4\) and \(b=14\).

Key Concepts

Multiplication of Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. The imaginary part includes the term with the imaginary unit, often symbolized by the letter \(i\). When multiplying complex numbers, it is essential to apply the distributive property, a key algebraic principle.
In multiplication, each part of one complex number multiplies with each part of the other. For the expression \(2i(7+2i)\), we need to follow this method by distributing \(2i\) to both terms inside the parentheses:

• First, multiply \(2i\) by the real number term \(7\).
• Then, multiply \(2i\) by the imaginary term \(2i\).

Throughout this process, remember to keep track of like terms, where terms containing \(i\) behave separately from purely real number terms. This approach ensures every component is multiplied correctly, leading to the appropriate expression where like terms can be combined later.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This definition allows us to solve equations that do not have solutions in the set of real numbers alone. The imaginary unit forms the basis for the imaginary part of complex numbers.
During multiplication involving the imaginary unit, it is crucial to apply the property of \(i\) correctly. In our example, the multiplication \(2i \cdot 2i\) demonstrates how the use of \(i^2 = -1\) turns what initially seems to be a complex result into a real number:

• Calculate \(2i \cdot 2i\) as \(4i^2\).
• Recognize that \(i^2\) equates to \(-1\), which transforms the multiplication into \(4(-1) = -4\).

This transformation simplifies the expression by eliminating \(i\) from the multiplication, converting it into a real number.

Distributive Property

The distributive property is a fundamental concept used to simplify expressions and solve equations. It states that \(a(b+c) = ab + ac\). This property allows us to distribute one quantity over a sum or difference within parentheses, providing a clear path to simplifying complex expressions.
When multiplying complex numbers, this property is particularly useful. It allows us to multiply each part of the complex number separately by the number outside the parentheses. In the example \(2i(7+2i)\), applying the distributive property:

• We first distribute \(2i\) to \(7\) (\(2i \cdot 7 = 14i\)).
• Next, distribute \(2i\) to \(2i\) (\(2i \cdot 2i = 4i^2 = -4\)).

By systematically applying the distributive property, complex multiplications are broken down into manageable parts. This allows us to simplify expressions in an organized manner, combining like terms to yield the final result suitable for interpretation and further calculation.