If $x$ is real and $x + \frac1x$ is rational, show by strong induction that $x^n + \frac{1}{x^n}$ is rational for all $n$. Suppose that $x$ subset of Real numbers such that $x + \frac{1}{x}$ is subset of Rational numbers. Using strong induction, show that for each $n$ subset of Natural numbers, $A_n = x^n + \frac{1}{x^n}$   is subset of Rational numbers. 
How do I start? I was given a hint of the product of $A_1$ and $A_n$ but have no idea how to apply.
thanks in advance!

If you need to use strong induction: Using the hint $$ A_1 A_n = x^{n+1}+\frac{1}{x^{n+1}} +x^{n-1}+\frac{1}{x^{n-1}} = A_{n+1} + A_{n-1}. $$ You want to rearrange this for an expression in $A_{n+1}$, the answer should follow from strong induction.
