Problem 56 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(-4 + 14i\).

Step 1: Apply the Distributive Property

To multiply the given complex expression, we will distribute the imaginary unit \(2i\) to each term within the parentheses \((7 + 2i)\). This means multiplying \(2i\) by \(7\) and then by \(2i\).

Step 2: Multiply \(2i\) by \(7\)

First, we multiply \(2i\) by \(7\): \(2i \times 7 = 14i\). This term represents the real number multiplied by the imaginary unit.

Step 3: Multiply \(2i\) by \(2i\)

Next, we multiply \(2i\) by \(2i\): \(2i \times 2i = 4i^2\). Remember that \(i^2 = -1\), so \(4i^2 = 4(-1) = -4\). This term is a real number.

Step 4: Combine Like Terms

Now, we combine the results from Step 2 and Step 3: \(14i + (-4)\). To write it in the form \(a + bi\), rewrite it as \(-4 + 14i\).

Key Concepts

Distributive Property

One of the foundational algebraic concepts is the distributive property. This property is vital in complex numbers multiplication. It tells us we can multiply a single term by every term inside a set of parentheses. For example, if you have a term outside the parentheses, you distribute it to each term inside. This means you effectively have a tiny multiplication task for each pair. In our original exercise, we see this with the expression:

• Multiplying the term outside the parentheses, which is 2i, with each term inside the parentheses, (7 + 2i).

This results in two separate operations:

• First, multiply 2i by 7.
• Second, multiply 2i by 2i.

This two-step process ensures that every part of the expression has been multiplied correctly before further simplification.

Imaginary Unit

The imaginary unit, represented as i, is key when dealing with complex numbers. The defining property of i is that it is the square root of -1, which means when squared, it gives -1. This concept can initially be confusing, but it's vital to handling expressions involving imaginary numbers. In complex multiplication, recognizing that:

• i^2 = -1 allows you to convert imaginary number operations into simpler forms that involve real numbers, like in the example where 4i^2 = 4(-1) = -4.

This transformation from imaginary to a negative real number is often critical in simplifying complex multiplication expressions.

Combining Like Terms

'Combining like terms' is a fundamental algebraic skill especially important when simplifying expressions. In the context of complex numbers, it involves grouping together real numbers and imaginary numbers after performing operations. For instance, in our exercise:

• After using the distributive property, you end up with terms like 14i and -4.

These terms are of different nature (one is imaginary, the other is real), yet they coexist in the form a + bi, where a is the real part and b the coefficient of i.
So, your final expression after combining like terms is arranged as

• -4 + 14i, representing the real and imaginary parts clearly.

This systematic approach ensures clarity and correctness in finalizing the solution.