Why do this problem?

The problem provides good practice in the manipuation of surds and in algebra (involving the expansions of $(p+q)^2$ and $(p+q)^3$, the difference of two squares and the use of the Remainder Theorem to factorise a cubic equation). If care is taken with the algebra the result comes out in a satisfyingly neat way. The question looks complicated but it turns out to be simple.

Possible approach
Although this is a longer, two part, question, the Hint gives sufficient guidance for this to be set to a class to work on independently.

Key questions

If 'extra' solutions are introduced by squaring or cubing, how do you decide which are the correct solutions?

This is how the problem was used by Peter Thomas, a Sixth Form College teacher:

I was absent at a meeting and set the class work to consolidate topics taught the previous lesson. The work was routine exercises from a textbook (Emanuel and Wood) which I encouraged them to approach selectively (what I called 'bread and butter' with some specific questions as a 'doggy bag' for homework).

Alongside this I set them the four nrich problems as 'cake' with the instruction to tackle at least one.

Ab Surd Ity (this problem)
System Speak
Root to Poly

Then over the following couple of weeks the problems were discussed in lessons as they related to the topics being covered.

The reaction was positive. The teacher covering the lesson reported strong engagement. From the individuals he mentioned this extended to those who liked to finish quickly then sit back. When the problems were discussed subsequently the contributions made suggested most had had a go and got somewhere with at least one problem.

It livened up what could have been a rather boring lesson of consolidation.