Square Root Of A Square Root Of A Square Root Is there some way to determine how many times one must root a number and its subsequent roots until it is equal to the square root of two or of the root of a number less than two?
sqrt(16)=4
sqrt(4)=2
sqrt(2) ... 3
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sqrt(27)=5.19615...
sqrt(5.19615...)=2.27950...
sqrt(2.27950...)=1.509803...
sqrt(1.509803...) ... 4
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Also, using the floor function...
sqrt(27)=floor(5.19615...)
sqrt(5)=floor(2.23606...)
sqrt(2) ... 3

Let's do this by example, and I'll let you generalize.
Say we want to know about $91$, one of my favorite numbers because it is the lowest number that I think most people might, at first thought, say is prime even though it isn't (another way of saying it isn't divisible by the 'easy-to-see' primes).
Well, I note that $16 = 2^4 < 91 < 256 = 2^8$
So if we take 2 square roots, then our number will be bigger than 2, as it's greater than $2^{4/4} = 2$. But If we take 3, as our number is less than $2^{8}$, then it's 3rd iterated square root will be less than $2^{8/8} = 2$.
So the third square root of $91$ will be between $\sqrt 2$ and $2$. So the 4th will place it below $\sqrt 2$.
