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Hi! This is my homework, and I want to finish it so I can have a nice and peaceful weekend.

There are 3 questions and 1 jumbo question, but they are not necessarily in the same category.

Please feel free to do them out of order, but please state what letter question it is!

(A) Question:  Please simplify: $$\frac{2\sqrt[3]{9}}{1\:+\:\sqrt[3]{3}\:+\:\sqrt[3]{9}}$$

(B) Question:  Find the solutions of the equation  $$x^2 + 36 = 0$$

(C) Question:  What are the roots of $$x^2 - 2x + 2?$$

JUMBO QUESTION!!!!!! *Please: show your work and put detail in this question.

There are six different sixth roots of 64. That is, there are six complex numbers that solve $$x^6=64$$ . Find them.

Thanks to the people who take time to answer questions!

CPhill and Alan usually answer my questions. Thanks to them!

Jan 26, 2019

#1
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(B): We subtract $$36$$ from both sides, leaving us with $$\sqrt{-36}=6i, -\sqrt{-36}=-6i$$ . Thus, the two solutions are $$\boxed{x=6i,\:x=-6i}.$$

(C): There are a few ways to solve this question:

Use $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$(thanks web2.0calc), and plugging in the values in the form $$ax^2+bx+c$$, we attain $$x=\frac{-\left(-2\right)+\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \:2}}{2\cdot \:1}=1+i$$ as one root, and $$x=\frac{-\left(-2\right)-\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \:2}}{2\cdot \:1}=1-i$$ . Thus, the two roots are $$\boxed{1+i, 1-i}.$$

Completing The Square:

Our main goal is to get it into the form $$x^2+2ax+a^2=\left(x+a\right)^2$$, so we have $$x^2-2x+\left(-1\right)^2=-2+\left(-1\right)^2$$, and simplifying, we get $$\left(x-1\right)^2=-1=x=i+1$$ and $$i-1$$ . Thus, the two roots are $$\boxed{1+i, 1-i}.$$

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Jan 26, 2019
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Nice job, tertre....!!!

CPhill  Jan 26, 2019
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Thank you! It took a long time to write the Latex!

tertre  Jan 26, 2019
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Tertre,

Your LaTex IS nice.  And it is great that you are learning and practicing your skills here.

PLUS

This person has clearly stated that they want someone to do their homework for them.

Why are you facilitating such blatent cheating?

It is not like this user can beat you up and take your lunch money if you do not help.

If that were the case I would understand better.

This is harsh, you are a great member and I don't want you to feel bad.

I'm sure I have at times been guilty of the same 'sin'.

I just don't want this to be known primarily as a free cheat site.

I want people to learn here, like you and I do.

Perhaps you could have part done it so that the user would at least have to finish the questions on their own.

Jan 26, 2019
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Correct, Melody, I forgot. I won't do it next time.

tertre  Jan 26, 2019
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Thanks Tertre.

Melody  Jan 26, 2019
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Can anyone do No. 1 or the jumbo question?

BIG THANKS TO TERTRE!!!

Jan 26, 2019
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JUMBO QUESTION

Using a circle in the complex plane  centered at (0,0) with a radius of 2 , the 6 roots of 64 are

2, 1 + √3 i , -1 + √3 i , -2, -1 - √3 i , 1 - √3 i

Here's a pic of the solutions

Jan 26, 2019
edited by CPhill  Jan 26, 2019