SOLUTION: {{{x=(3 + sqrt ( 5 ))/(3 - sqrt ( 5 ))}}} and {{{y =(3 - sqrt ( 5 ))/(3 + sqrt ( 5 ))}}}. calculate {{{x^3 + y^3}}}

Algebra ->  Equations -> SOLUTION: {{{x=(3 + sqrt ( 5 ))/(3 - sqrt ( 5 ))}}} and {{{y =(3 - sqrt ( 5 ))/(3 + sqrt ( 5 ))}}}. calculate {{{x^3 + y^3}}}      Log On


   

Question 762576: x=%283+%2B+sqrt+%28+5+%29%29%2F%283+-+sqrt+%28+5+%29%29 and y+=%283+-+sqrt+%28+5+%29%29%2F%283+%2B+sqrt+%28+5+%29%29. calculate x%5E3+%2B+y%5E3
Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
This a lot of multiplying, but here goes. Start with
(1) x+=+%283+%2B+sqrt%285%29%29%2F%283+-+sqrt%285%29%29
Now multiply numerator and denominator of (1) by
(2) %283+%2B+sqrt%285%29%29 and get
(3) x+=+%289+%2B+6sqrt%285%29+%2B5%29%2F4 or
(4) x+=+%2814+%2B+6sqrt%285%29%29%2F4 or
(5) x+=+%287+%2B+3sqrt%285%29%29%2F2
Now get
(6) x%5E2+=+%2849%2B42sqrt%285%29%2B45%29%2F4 or
(7) x%5E2+=+%2894%2B42sqrt%285%29%29%2F4
Now get x%5E3 from
(8) x%5E3+=+x%2Ax%5E2 or
(9) x%5E3+=+%287+%2B+3sqrt%285%29%29%2A%2894%2B42sqrt%285%29%29%2F8 or
(10) x%5E3+=+%287%2A94+%2B+%283%2A94+%2B+7%2A42%29%2A%28sqrt%285%29%29%2B+3%2A42%2A5%29%2F8 or
(11) x%5E3+=+%28658+%2B+576%28sqrt%285%29%29%2B+630%29%2F8 or
(12) x%5E3+=+161+%2B+72sqrt%285%29
From the original problem we see that
(13) y = 1/x or
(14) y%5E3+=+1%2Fx%5E3 or
(15) y%5E3+=+1%2F%28161%2B72sqrt%285%29%29
Now add (12) and (15) to get the desired sum
(16) x%5E3%2By%5E3=+161+%2B+72sqrt%285%29+%2B++1%2F%28161%2B72sqrt%285%29%29 or
(17) or
(18) x%5E3%2By%5E3=+%2851842+%2B+23184sqrt%285%29%29%2F%28161%2B72sqrt%285%29%29
The GCF of the numerator of (18) is 2*7*23 or 322. So we can write (18) as
(19) x%5E3%2By%5E3=+322%2A%28161+%2B+72sqrt%285%29%29%2F%28161%2B72sqrt%285%29%29 which reduces to
the final answer,
(20) x%5E3%2By%5E3=+322