Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$-3 i(6-i)$$

Step-by-Step Solution

Answer

The answer is \(-3 - 18i\)

Step 1: Distribute -3i into (6 - i)

Let's distribute the number -3i into the parentheses. This means multiplying -3i by each term within the parentheses, namely 6 and -i.

Step 2: Multiply -3i by 6 and -i

The multiplication gives us the following: \(-3i \cdot 6 = -18i\), \(-3i \cdot -i = 3i^2\). The term \(i^2\) is equal to -1 so the result is 3*(-1) = -3.

Step 3: Sum up the results

Combine the results of the multiplication. This gives \(-18i - 3\). But the standard form for complex numbers is a + bi, so we need to rearrange it to \(-3 - 18i\).

Key Concepts

Distributing Complex Numbers

Understanding how to distribute complex numbers is crucial when simplifying expressions that involve them. Similar to distributing real numbers, distributing a complex number involves multiplying the number outside the parentheses by each term within the parentheses.

For example, with the expression \( -3i(6-i) \), you multiply \( -3i \) by each of the terms inside the parentheses: 6 and \( -i \). This is the initial step to break down our complex number expression into simpler parts before we add or subtract those resulting terms to find the standard form of the complex number.

Multiplying Imaginary Numbers

• Remember that the imaginary unit \( i \) is defined such that \( i^2 = -1 \).
• When you multiply an imaginary number by itself, you're dealing with a power of \( i \). Specifically, for \( i^2 \), you will substitute \( -1 \) for \( i^2 \), turning the expression into a real number.
• In our example, \( -3i \) times \( -i \) gives \( 3i^2 \) which simplifies to \( 3(-1) \), or \( -3 \) once you replace \( i^2 \) with \( -1 \).

By applying the property of \( i^2 \) consistently whenever you multiply imaginary numbers, you'll be able to simplify complex products with ease.

Complex Number Operations

After distributing and multiplying, you often end up with two kinds of terms: real numbers and imaginary numbers (terms that include the imaginary unit \( i \)). Complex number operations include adding, subtracting, multiplying, and dividing both types of these terms to reach a standard form of a complex number, which is expressed as \( a + bi \), where \(a \) and \(b \) are real numbers.

In the provided exercise, we have \( -18i \) and \( -3 \) after distributing and multiplying. The next operation is to combine these results. The correct standard form is \( -3 - 18i \) where \( -3 \) is the real part and \( -18i \) is the imaginary part. Always remember to express your final answer with the real part first, followed by the imaginary part.