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Chapter 7: Problem 36

Multiply and simplify. Write each answer in the form \(a+b i\). $$ 3 i \cdot 9 i $$

Short Answer

Expert verified
The simplified answer is \( -27 \)

Step by step solution

01

Multiply the coefficients

First, multiply the coefficients of the complex numbers. Here, the coefficients are 3 and 9. The multiplication of these coefficients is given by: \[ 3 \times 9 = 27 \]
02

Multiply the imaginary units

Next, multiply the imaginary units ( \( i \)). When multiplying two imaginary units, \( i \times i \) gives \( i^2 \). According to the properties of imaginary numbers, \( i^2 = -1 \). So: \[ i \times i = i^2 = -1 \]
03

Combine the results

Combine the results from Step 1 and Step 2. Multiplying the 27 by the value of \( i^2 \): \[ 27 \times (-1) = -27 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Numbers
Imaginary numbers are numbers that involve the imaginary unit, often denoted as \(i\). The imaginary unit \(i\) is defined as the square root of -1. This means:
\[i = \sqrt{-1} \]
Imaginary numbers are useful in a wide range of mathematical applications, from solving quadratic equations to engineering. Anytime you see \(i\), you know you are working with imaginary numbers.
Complex Number Multiplication
Multiplying complex numbers involves a few key steps that are simple once you get the hang of it.
  • Step 1: Multiply the coefficients. In our example, the coefficients are 3 and 9. Multiplying them gives:
    \[ 3 \times 9 = 27 \]
  • Step 2: Multiply the imaginary units. Imaginary units follow specific rules. When multiplying two \(i\)'s together:
    \[ i \times i = i^2 = -1 \]
  • Step 3: Combine the results. Multiply your result from the coefficients by \(i^2\):
    28
    \[ 27 \times (-1) = -27 \]

These steps will help you multiply complex numbers and simplify them correctly.
Simplifying Complex Expressions
Simplifying complex expressions often involves turning complex multiplications and additions into their simplest form.
  • Step 1: Perform all the multiplications, including both coefficients and imaginary units.
  • Step 2: Use the definition \(i^2 = -1\) to simplify any \(i^2\) terms.
  • Step 3: Combine the simplified coefficients to reach the final answer.

For example, if you multiply 3i by 9i, follow the steps to get:

\[ 3i \times 9i = 27i^2 = 27(-1) = -27 \]
Simplifying complex expressions sometimes seems tricky, but following the steps methodically makes it straightforward.

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Most popular questions from this chapter

Describe a procedure that uses the distance formula to determine whether three points, \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right),\) are vertices of a right triangle.

Let \(f(x)=x^{2} .\) Find each of the following. Find the slope and the \(y\) -intercept of the line given by \(3 y+5 x=1\)

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|-3-i|$$

A cube measures \(5 \mathrm{cm}\) on each side. How long is the diagonal that connects two opposite corners of the cube? Give an exact answer. (Image can't copy)

Simplify. $$5 i^{5}+4 i^{3}$$
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