/*! 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 92 Cube each complex number. (a) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 92

Cube each complex number. (a) \(-1+\sqrt{3} i\) (b) \(-1-\sqrt{3} i\)

Short Answer

Expert verified
(a) Cube of \(-1+\sqrt{3} i\) is \(1-\sqrt{3}i\), (b) Cube of \(-1-\sqrt{3} i\) is \(1+\sqrt{3}i\).

Step by step solution

01

Cubing the first Complex Number

Start with the first complex number \(-1 + \sqrt{3} i\). To cube this, replicate it three times and multiply: \[ (-1 + \sqrt{3} i)^3 = (-1 + \sqrt{3} i)(-1 + \sqrt{3} i)(-1 + \sqrt{3} i)\] Using the FOIL method (i.e., multiplying the 'First' terms, the 'Outer' terms, the 'Inner' terms, and the 'Last' terms), and recalling that \(i^2 = -1\), simplify the equation.
02

Calculating the First Cube

Simplify the above equation: \[(-1 + \sqrt{3} i)^3 = -1 - \sqrt{3} i - \sqrt{3} i +3 -1 +\sqrt{3} i + \sqrt{3} i -3 = -1 - \sqrt{3} i +2 = 1 - \sqrt{3} i . \]
03

Cubing the Second Complex Number

Repeat the process with the second complex number \(-1 - \sqrt{3} i\): \[(-1 - \sqrt{3} i)^3 = (-1 - \sqrt{3} i)(-1 - \sqrt{3} i)(-1 - \sqrt{3} i).\] Again, utilize the FOIL method and \(i^2 = -1\) to simplify.
04

Calculating the Second Cube

Simplify the above equation: \[(-1 - \sqrt{3} i)^3 = -1 +\sqrt{3} i - \sqrt{3} i + 3 -1 - \sqrt{3} i - \sqrt{3} i + 3 = -1 + \sqrt{3} i + 2 = 1 + \sqrt{3} i. \]

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!

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

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{3}-x^{2}+x+39$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$2 x^{2}+b x+5=0$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.