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1)

 

Simplify \(\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.\)

2)

 

What are the roots of x^2 - 2x + 2?

 

3)

 

Express (4-5i)(-5+5i) in the form a+bi, where a and b are integers and i=\sqrt{-1} is the imaginary unit.

 Feb 2, 2020
 #1
avatar+107466 
0

 

\(\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}\\~\\ =\frac{2*9^{\frac{1}{3}}}{1 +3^{\frac{1}{3}} +9^{\frac{1}{3}}}\\~\\ =\frac{2*3^{\frac{2}{3}}}{1 +3^{\frac{1}{3}} +3^{\frac{2}{3}}}\\~\\ let\;\; x=3^{1/3}\\~\\ =\frac{2x^2}{1 +x +x^2}\\~\\ =\frac{2x^2}{x^2+x+1}\\~\\ \)

I really do not see this simplifying very nicely

 

 

coding:

\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}\\~\\ 
=\frac{2*9^{\frac{1}{3}}}{1 +3^{\frac{1}{3}} +9^{\frac{1}{3}}}\\~\\
 =\frac{2*3^{\frac{2}{3}}}{1 +3^{\frac{1}{3}} +3^{\frac{2}{3}}}\\~\\ 
let\;\; x=3^{1/3}\\~\\ 
=\frac{2x^2}{1 +x +x^2}\\~\\ 
=\frac{2x^2}{x^2+x+1}\\~\\ 
 

 Feb 2, 2020
edited by Melody  Feb 2, 2020
 #2
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+1

Hi Melody: See this elegant answer # 11 by Max Wong:  https://web2.0calc.com/questions/help-plzzzz_3

 Feb 2, 2020
 #3
avatar+107466 
0

Thanks guest and thanks to Max.

Max's answer required very good visual insight. 

I always knew he was clever from when he first started posting.

 

Unfortunately I am not likely to remember this.  How sad.

Melody  Feb 2, 2020

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