If $(w + 1)(w - 1) = w$, find $ { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } } $. Recently I was asked a question by my student that completely stumped me.

$$\text{If }(w + 1)(w - 1) = w\text{, find } { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } }. $$

One "cheat" method that I used was to solve for the exact value of $w$ from the given first equation, and then substitute it into the requested expression that we were asked to find. I got $123$ as the answer.
However, I'm quite sure there's an algebraic way to solve this. 
Anyone wants to give this a ahot?

HINT:
If $F_n=w_1^n+w_2^n$ where $w_1,w_2$ are the roots of $w^2-w-1=0$
$F_0=1+1,F_1=w_1+w_2=1$
$F_{n+2}=F_{n+1}+F_n$
$n=0\implies F_{0+2}=F_{0+1}+F_0=2+1$
$F_{1+2}=F_{1+1}+F_1=3+2$
Now $w_1^{2m}+\dfrac1{w_1^{2m}}=w_1^{2m}+w_2^{2m}=?$
