If there's a specific algorithm to calculate roots (√) on calculators, why can't we just derive a formula from that algorithm? This just cross my mind earlier today. There's/are computer algorithms (the code used by the calculator) to calculate any root of any number, but there's still no definite way to get any root of any number.
 A: There is a way to do that.
If you want to calculate $\sqrt a$, you can calculate the terms of the sequence
$$x_{n+1}=\frac 12\left(x_n+\frac a{x_n}\right)$$
and $x_0=1$.
It will give you quite quickly a nice approximation of $\sqrt a$.
A: See Newton's Method. We definitely have algorithms but it's too time consuming to use them without computers. We can find the $n^{th}$ root using Newton's method. There are several other methods too including the binomial theorem. You just need to do adequate research yourself before asking a question. It will be more interesting then. 
A: The word "algorithm" in your question has at least two possible meanings. You might want a sequence of arithmetic operations that provides the answer in a finite number of steps - like the algorithms you learn in grade school for multiplication or division for integers.
There is no such algorithm for roots of numbers when those roots are irrational. But there are algorithms like the one in @EJoseph 's excellent answer that provide any prespecified level of precision for an approximation to the root. Your calculator uses one of them. The required level of precision is the number of digits in the display.  
A: Numerical methods are indeed how calculators produce the approximations of square roots that they report to the user. However, they are just that, approximations; they are not the exact value. The relevant series that can be made from most of these methods are exact representations, but they are not closed form (i.e. they are infinite sums). The notation $\sqrt{x}$, is, by definition, an expression for the square root of $x$ and is also closed form (if we permit this symbol to appear in what we will call closed-form expressions; the notion of being closed-form only makes sense relative to some specific 'vocabulary' of symbols with assigned meanings, otherwise we can always just invent a new symbol for what we want to represent to give it a closed-form expression).
