/*! 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 6 Express each number in terms of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 6

Express each number in terms of i. $$\sqrt{-12}$$

Short Answer

Expert verified
The solution is \(2i\sqrt{3}\).

Step by step solution

01

Express Negative Root in terms of 'i'

We can write \(\sqrt{-12}\) as \(\sqrt{-1 * 12}\). The square root of -1 is \(i\) so it becomes \(i\sqrt{12}\).
02

Simplify Root

\(\sqrt{12}\) can be simplified to \(2\sqrt{3}\) so it simplifies to \(i * 2\sqrt{3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Imaginary Unit
When dealing with complex numbers, the imaginary unit, denoted as \(i\), plays a central role. It is defined as the square root of -1.

This definition is crucial because it allows us to extend the concept of square roots to negative numbers, which we otherwise cannot do with real numbers alone.

For example:
  • \(i^2 = -1\), which follows directly from its definition.
  • Because \(i = \sqrt{-1}\), any time you encounter the square root of a negative number, it generally involves \(i\).
Understanding \(i\) as an imaginary unit helps us bridge the gap between real numbers and complex numbers, paving the way for solving a wider array of mathematical problems.
Simplifying Radicals
Simplifying radicals is an important skill when working with both real and complex numbers. The goal is to express the radical in its simplest form.

To simplify \( \sqrt{12} \), we first look for the largest perfect square that divides evenly into the number under the square root.

Steps for simplifying \( \sqrt{12} \):
  • Identify the factors of 12: 1, 2, 3, 4, 6, 12.
  • The largest perfect square is 4, and \(4 \times 3 = 12\).

Now, rewrite \( \sqrt{12} \) as \( \sqrt{4 \times 3} \). Since \(\sqrt{4} = 2\), you can simplify \( \sqrt{12} \) to \(2\sqrt{3}\).

Simplifying radicals helps make expressions easier to work with, especially when combined with imaginary numbers.
Square Roots of Negative Numbers
Square roots of negative numbers can seem challenging at first, but with the help of the imaginary unit \(i\), they become manageable.

When you encounter a square root of a negative number, the key is to first isolate the negative part by using \(i\).

For example:
  • Consider \( \sqrt{-12} \).
  • Start by rewriting it as \( \sqrt{-1 \cdot 12} \).
  • We know that \( \sqrt{-1} = i \), so replace \(\sqrt{-1}\) with \(i\), giving you \(i\sqrt{12}\).
  • Finally, simplify \(\sqrt{12}\) to \(2\sqrt{3}\) to get \(i \cdot 2\sqrt{3}\).

By using \(i\), you effectively bring the negative component outside of the square root, allowing you to handle it just like any other radical expression.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the parabola. $$y=-4 x^{2}+20 x+160$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because I want to solve \(25 x^{2}-49=0\) fairly quickly, I'll use the quadratic formula.

Solve: $$x^{2}+2 \sqrt{3} x-9=0$$

Llist the numbers from each set that are: (A). rational numbers; (B). irrational numbers; (C). real numbers; (D). not real numbers. (Hint: Your answer to each question in Exercise 85 should be "no." $$[-\sqrt{9},-\sqrt{7}, \sqrt{-9}, \sqrt{-7}, \sqrt{0}, \sqrt{7}, \sqrt{9}]$$

Use the formula for the area of a circle, \(A=\pi r^{2},\) to solve Exercises \(71-72\). If the area of a circle is \(36 \pi\) square inches, find its radius.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.