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?

From $w^2=w-1=0$ by dividing with $w$, we get
$$w-\frac{1}{w}=1,$$ from where we get $$w^2+\frac{1}{w^2}=(w-\frac{1}{w})^2+2=3.$$
Similarly, we have $$w^4+\frac{1}{w^4}=(w^2+\frac{1}{w^2})^2-2=7$$
and $$w^8+\frac{1}{w^8}=47.$$
Since $$w^{10}+\frac{1}{w^{10}}=(w^2+\frac{1}{w^2})(w^8+\frac{1}{w^8})-(w^2+\frac{1}{w^2})(w^4+\frac{1}{w^4}-1),$$
we have $$w^{10}+\frac{1}{w^{10}}=3\times 47-3\times (7-1)=3\times 41 =123$$  
