SOLUTION: If (x+(1/x))^2 = 7 and x is a positive real number, find the exact value of x^3 + 1/x^3 Thank you!

Algebra ->  Expressions -> SOLUTION: If (x+(1/x))^2 = 7 and x is a positive real number, find the exact value of x^3 + 1/x^3 Thank you!      Log On


   

Question 1086530: If (x+(1/x))^2 = 7 and x is a positive real number, find the exact value of x^3 + 1/x^3
Thank you!
Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  From  %28x%2B%281%2Fx%29%29%5E2 = 7  you have  x%2B%281%2Fx%29 = sqrt%287%29.


2.  It implies 

    %28sqrt%287%29%29%5E3 = 7%2Asqrt%287%29 = %28x+%2B+1%2Fx%29%5E3 = x%5E3+%2B+3x%5E2%2A%281%2Fx%29 + 3x%2A%281%2Fx%29%5E2+%2B+%281%2Fx%29%5E3 = x%5E3+%2B+1%2Fx%5E3 + 3x+%2B+3%2A%281%2Fx%29 = %28x%5E3+%2B+1%2Fx%5E3%29 + 3%2A%28x+%2B+1%2Fx%29.


    Now replace in the last term  x+%2B+1%2Fx  by  sqrt%287%29  (based on n.1),  and you will get

    7%2Asqrt%287%29 = %28x%5E3+%2B+1%2Fx%5E3%29 + 3%2Asqrt%287%29,   or


    x%5E3+%2B+1%2Fx%5E3 = 4%2Asqrt%287%29.

Answer.   x%5E3+%2B+1%2Fx%5E3 = 4%2Asqrt%287%29.


Solved.