<s>[INST] Why am I getting two answers for 8th root of continued fraction Find value of $x$:
$x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$
On solving ,we have $x^8=2207-\frac{1}{x^8}$
$x^8+\frac{1}{x^8}=2207$
$x^4+\frac{1}{x^4}=47$
$x^2+\frac{1}{x^2}=7$
$x+\frac{1}{x}=3$
$x=\frac{3+\sqrt{5}}{2}\,,\frac{3-\sqrt{5}}{2}$
But $x$ can take only one value. 
The problem is which of these two values is to be accepted/rejected and why.
 [/INST] You want the value which is greater than $1$ - clearly by estimating.
From $x^8+\frac 1{x^8}=2207$ you know that if $x$ satisfies that equation so will $\frac 1x$. Put $\frac 1x$ in the original equation and modify the eighth root accordingly and you will see where the alternative answer comes from and that this is clearly less than $1$.
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