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Simplify $\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.$ Thanks to anyone who helps!

 Jun 27, 2019
 #1
avatar+104962 
+3

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

 

Note that

 

(  1)^3 - ( ∛3)^3  =   ( 1 - ∛3 ) ( 1 + ∛3 + (∛3)^2)  =     ( 1 - ∛3 )   ( 1 - ∛3 + ∛9)

 

So  this implies that if we multiply the numerator and  denominator by  ( 1 - ∛3)  we get

 

2∛9 ( 1 - ∛3)                         2 ∛9 - 2∛9 *∛3        2∛9 - 2∛27          2 [ ∛9 - 3 ]

__________________  = ______________  =  ___________  = ___________ =

( 1 - ∛3 + ∛9) ( 1 - ∛3)         (1)^3  - (∛3)^3            1  -  3                    -2

 

 

- [ ∛9 - 3 ]     =

 

3 - ∛9      [  =   3  - 3^(2/3) ]

 

 

cool cool cool

 Jun 27, 2019
 #2
avatar
+2

See other solutions here, particularly answer # 11.

 

https://web2.0calc.com/questions/help-plzzzz_3

 Jun 27, 2019

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