SOLUTION: Please help me solving it If {{{a=(1/2) (5-21^(1/2))}}} then the value of {{{a^3+a^(-3)-5a^2-5a^(-2)+a+a^(-1)}}}is?

Algebra ->  Exponents -> SOLUTION: Please help me solving it If {{{a=(1/2) (5-21^(1/2))}}} then the value of {{{a^3+a^(-3)-5a^2-5a^(-2)+a+a^(-1)}}}is?      Log On


   

Question 1046033: Please help me solving it
If a=%281%2F2%29+%285-21%5E%281%2F2%29%29 then the value of a%5E3%2Ba%5E%28-3%29-5a%5E2-5a%5E%28-2%29%2Ba%2Ba%5E%28-1%29is?
Answer by ikleyn(52884) About Me  (Show Source):
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Please help me solving it
If a=%281%2F2%29+%285-21%5E%281%2F2%29%29 then the value of a%5E3%2Ba%5E%28-3%29-5a%5E2-5a%5E%28-2%29%2Ba%2Ba%5E%28-1%29 is ?
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First, let us prove this statement: 


    if  a = %281%2F2%29%2A%285-sqrt%2821%29%29  then  a%5E%28-1%29 = 1%2Fa = %281%2F2%29%2A%285%2Bsqrt%2821%29%29.


Indeed,  %281%2F2%29%2A%285-sqrt%2821%29%29*%281%2F2%29%2A%285%2Bsqrt%2821%29%29 = %281%2F4%29%2A%285%5E2+-+%28sqrt%2821%29%5E2%29%29 = %281%2F4%29%2A%2825-21%29 = %281%2F4%29%2A4 = 1.


OK. So, if  a = %281%2F2%29%2A%285-sqrt%2821%29%29,  then  a+%2B+a%5E%28-1%29 = %281%2F2%29%2A%285-sqrt%2821%29+%2B+5+%2B+sqrt%2821%29%29 = 10%2F2 = 5.


Having this, we can make the next step.

a%5E2 + 1%2Fa%5E2 = %28a+%2B+1%2Fa%29%5E2+-+2 = 5%5E2+-+2 = 23.   (I replaced here  %28a+%2B+1%2Fa%29  by 5, based on the proof above )


Furthermore, 

a%5E3 + 1%2Fa%5E3 = %28a%2B+1%2Fa%29%2A%28a%5E2+-+1+%2B+1%2Fa%5E2%29          ( I applied the formula x%5E3+%2B+y%5E3 = %28x%2By%29%2A%28x%5E2-xy%2By%5E2%29

            = 5%2A%2823-1%29 = 110.             ( I replaced here a+%2B+1%2Fa by 5  and replaced  a%5E2+-+1 + 1%2Fa%5E2%29  by (23-1) = 22 based on results proved above).

Thus we have  a+%2B+1%2Fa = 5,  a%5E2 + 1%2Fa%5E2 = 23  and  a%5E3 + 1%2Fa%5E3 = 110.


Then the value under the question is  110 - 5*23 + 5 = 0.

For similar problems, see the lesson
    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29  and  %28x%5E3%2B1%2Fx%5E3%29
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.