Problem 53 - Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i.\) $$ 2 i(7-3 i) $$

Step-by-Step Solution

Answer

The answer is \(6 + 14i\).

Step 1: Distribute the Terms

To multiply the term outside the parentheses by each term inside, distribute the multiplication as follows: \[ 2i imes 7 + 2i imes (-3i) \]

Step 2: Simplify Each Product

Calculate each product separately:1. \(2i imes 7 = 14i\)2. \(2i imes (-3i) = -6i^2 \)Recall that \( i^2 = -1 \), therefore, \(-6i^2 = 6\).

Step 3: Combine Real and Imaginary Parts

Add the real and imaginary components from the simplified products:\[ 6 + 14i \]This is the answer in the form \( a + bi \).

Key Concepts

Multiplying Complex Numbers

Complex numbers can be a bit tricky at first, but with a bit of practice, you'll get the hang of it! Multiplying them might seem different from multiplying regular numbers, but it's quite straightforward when you break it down step by step. In the original exercise, you're asked to multiply

• 2i by the binomial expression (7 - 3i).

The solution involves taking the number outside the parentheses, which is 2i, and multiplying it by each term inside the parentheses. This method is similar to what you would do in algebra with regular numbers, ensuring every term in the binomial is multiplied by the term outside. Remember, the goal is to rewrite the multiplication in the form

• a + bi, where a and b are real numbers.

The imaginary part of the result will have the i unit, which is the imaginary unit itself. Once you distribute properly, you're already halfway through!

Distributive Property

The distributive property is a cornerstone in algebra, and it applies to complex numbers just like it does for real numbers. Basically, it lets us distribute the multiplication of one number to each term inside a set of parentheses. It's like a magician spreading magic equally to everything in their hat!
In the original problem, we started with 2i (7 - 3i). To solve it, we multiplied 2i by each term within the parentheses:

• 2i * 7 yields 14i
• 2i * (-3i) gives us -6i²

Notice here that we've correctly applied the distributive property to each component of the binomial. Solving these one by one and then putting it all together in the form

• a + bi makes it all add up perfectly!
  Just remember this magical principle and you’ll be able to take on more complex challenges with confidence.

Imaginary Unit

The imaginary unit, denoted as i, is a fundamental part of complex numbers. It's what distinguishes them from real numbers. The special thing about i is that it represents the square root of -1. And, like magic, it follows the rule that i² = -1. This characteristic is essential when you're working on multiplying complex numbers.
Going back to our example from the exercise, during Step 2, you ended up with a term -6i².
Here’s where the trick with the imaginary unit comes into play!

• Since i² = -1, this means that -6i² becomes 6.

This conversion allowed us to simplify and resolve the equation into the form

• a + bi, yielding 6 + 14i.

Understanding how i behaves is crucial as it ensures that we can translate between these unfamiliar-sounding concepts into something workable and familiar.