x to the power 1029 + 1/x to the power 1029 =3 then find x

Question

x¹⁰²⁹ + 1/x¹⁰²⁹ = 3 then x¹⁰²⁹ - 1/x =?

Answer 1

I wonder whether you have conveyed the original question question correctly but here goes.

 

x¹⁰²⁹ + 1/x¹⁰²⁹ = 3

 

 

Substitute a=x¹⁰²⁹ 

 

a + 1/a =3

 

Multiplying all terms  by a and collect terms to one side gives

 

a² -3a +1 =0

 

Solve the quadratic for a gives

 

a= (3±√5)/2

 

a=2.618033989... (which happens to be the golden mean +1)

 

and 

 

a=0.381966011...

 

 

Solving for x

 

x¹⁰²⁹=a

 

log(x¹⁰²⁹)=log(a)

 

1029 log(x)=log(a)

 

log(x)=log(a)/1029

 

x=10^{(log(a)/1029)} 

 

For a =0.38196601...  

 

x=0.999065137

 

For a = 2.618033989...

 

x=1.000935737

 

Note the two x solutions are reciprocals of each other.

 

So for x=0.999065137

 

x¹⁰²⁹=0.381966011

 

Then

x¹⁰²⁹ -1/x = 1.382901749

 

And for x=1.000935737

 

x¹⁰²⁹=2.618033989...

 

Then
x¹⁰²⁹ -1/x = 3.617099126