Problem 49 - Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i.\) $$ 3(2-9 i) $$

Step-by-Step Solution

Answer

The result of multiplying is \(6 - 27i\).

Step 1: Apply the distributive property

To multiply the expression, we need to distribute the real number 3 across the complex number \((2 - 9i)\). This means multiplying 3 with each term inside the parentheses.

Step 2: Multiply the real part

First, multiply the real number part: \(3 \times 2 = 6\).

Step 3: Multiply the imaginary part

Next, multiply the imaginary part: \(3 \times (-9i) = -27i\).

Step 4: Combine the results

Combine the results from Step 2 and Step 3 to form the new complex number: \(6 - 27i\).

Key Concepts

Distributive Property

The distributive property is a valuable tool when working with algebraic expressions, including those involving complex numbers. It refers to the way we multiply a single term by each term inside a set of parentheses. When you see something like

• \(a(b+c)\),
• you can distribute the \(a\) to get \(ab + ac\).

In the context of the original exercise with complex numbers, the distributive property allows us to handle the multiplication of a real number by each part (real and imaginary) of the complex number separately. When applied to

• \(3(2-9i)\),
• it breaks down into two separate operations: multiplying 3 by 2 and multiplying 3 by \(-9i\).

This stepwise multiplication ensures all parts of both numbers are appropriately combined.

Real and Imaginary Parts

Complex numbers are composed of two distinct parts: the real and the imaginary. In expressions like

• \(a + bi\),
• \(a\) is the real part,
• and \(bi\) is the imaginary part.

The imaginary unit, denoted as \(i\), is defined as the square root of \(-1\). In calculations involving complex numbers, both parts need to be addressed separately.
In the original example \(3(2-9i)\),

• \(2\) is the real part,
• and \(-9i\) is the imaginary part.

By understanding these components, you can easily apply arithmetic operations and combine them as required.
Thus, when you're multiplying or adding complex numbers, always remember to handle each part explicitly, ensuring clear results.

Multiplying Complex Numbers

Multiplying complex numbers might initially seem daunting but is straightforward once you understand the pattern. This involves applying the distributive property, allowing each component of the first number to be multiplied by each component of the second. In the form of

• \((a + bi)(c + di)\),
• you calculate it as \(ac + adi + bci + bdi^2\).

Note that \(i^2\) is equal to \(-1\), so wherever \(i^2\) appears, it should be replaced accordingly.
In our example

• \(3(2-9i)\),
• you multiply 3 by both the real and imaginary components separately,
• producing \(6\) and \(-27i\) respectively.

These values combine to give the final result,

• \(6 - 27i\).

This straightforward multiplication process ensures you can work with any complex numbers with confidence, reinforcing clarity in handling both real and imaginary parts.