TITLE: Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$
QUESTION [8 upvotes]: A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it?
$$\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x = 1$$
I'm seeing a relation between $4x$, $16x$ and $64x$ so maybe the larger can be simplified to the smaller?
I do encourage you to working within an examination environment (thus not making use of anything other than pen and paper and possibly a calculator). 

EDIT: My question had a missing $-\sqrt x$ at the end, sorry!

REPLY [8 votes]: This is a very intuitive approach based on the fact that I assume to be in the exam room with no computer and even no calculator. 
If there is a simple root $\sqrt{64x+5}$ should reduce to a whole number and $x=\frac1{16}$ is obvious (since $9$ is the closest square to $5$). From here, we can go backward (verify that every time we get another square) and check that this is the solution.