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The following important properties of the square root functions are valid for all positive réal numbers x and y:

for every real number x (see absolute value)

Fungsi akar kuadrat umumna metakeun wilangan rasional ka wilangan aljabar; √x is rational if and only if x is a rational number which, after cancelling, is a fraction of two perfect squares. In particular, √2 is irrational.

In geometrical terms, the square root function maps the area of a square to its side length.

Suppose that x and a are réals, and that x2=a, and we want to find x. A common mistake is to "take the square root" and deduce that x = √a. This is incorrect, because the square root of x2 is not x, but the absolute value |x|, one of our above rules. Thus, all we can conclude is that |x| = √a, or equivalently x = ±√a.

In calculus, for instance when proving that the square root function is continuous or differentiable or when computing certain limits, the following identity often comes handy:

It is valid for all non-negative numbers x and y which are not both zero.

The function f(x) = √x has the following graph, made up of half a parabola lying on its side:

The function is continuous for all non-negative x, and differentiable for all positive x (it is not differentiable for x=0 since the slope of the tangent there is ∞). Its derivative is given by

Its Taylor series about x = 1 can be found using the binomial theorem: