/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Find each product. $$ (-6 i)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 21

Find each product. $$ (-6 i)(-i) $$

Short Answer

Expert verified
The product is -6.

Step by step solution

01

- Identify the components

The problem involves multiplying two complex numbers: \(-6i\) and \(-i\).
02

- Multiply the coefficients

First, multiply the coefficients of \(-6i\) and \(-i\). The coefficients are -6 and -1, respectively. \(-6 \times -1 = 6\).
03

- Multiply the imaginary units

Next, multiply the imaginary units \i\ and \i\. This gives \(i \times i = i^2\).
04

- Simplify imaginary unit multiplication

Recall that \(i^2 = -1\). Therefore, \(6i^2 = 6 \times -1 = -6\).
05

- Combine the results

Combine the results from the previous steps to get the final product.

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Key Concepts

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

complex numbers
Complex numbers are numbers that consist of a real part and an imaginary part. A standard complex number is written in the form of: \(a + bi\), where \(a\) is the real part, and \(bi\) represents the imaginary part.
A key feature of complex numbers is how they interact with each other when added, subtracted, multiplied, or divided.
For example, in this exercise, we multiplied two imaginary terms \(-6i\) and \(-i\). Both of these can be considered complex numbers with the real part being zero.
Understanding complex numbers allows us to solve a broader range of mathematical problems that cannot be solved using only real numbers.
imaginary unit
The imaginary unit is symbolized as \(i\), and it has a unique property: \(i^2 = -1\).
This property is fundamental to complex numbers.
When we see \(i\), it represents the square root of \(-1\).
Despite seeming strange at first, the imaginary unit provides a way to extend the number system beyond the real numbers.
In this exercise, when we multiplied \(-6i\) and \(-i\), the term \(i\) appeared twice. According to \(i\)'s property: \(i \times i = i^2\). Since \(i^2 = -1\), this helped us transform the expression.
properties of i
The properties of \(i\) are crucial to working with complex numbers.
These properties include:
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

These properties repeat cyclically.
In the exercise, after determining that \(i \times i = i^2\), we used the property \(i^2 = -1\) to simplify the expression.
Recognizing these repeating patterns is vital for handling more complex calculations involving imaginary numbers.
In summary, understanding and leveraging the properties of \(i\) simplifies the operations on complex numbers and imaginary units.

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Most popular questions from this chapter

Find each product. $$ (11-2 i)(11+2 i) $$

Concept Check Which step is an appropriate way to begin solving the quadratic cquation \(2 x^{2}-4 x=9\) by completing the square? A. Add 4 to each side of the equation. B. Factor the left side as \(2 x(x-2)\) C. Factor the left side as \(x(2 x-4)\) D. Divide each side by 2

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ 5 x^{2}=3-x $$

Evaluate each expression for \(x=3\). $$ 2 x^{2}-x+1 $$

Solve equation by using the square root property. Simplify all radicals. \((3 k+1)^{2}=18\)
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