Radical

I am not yet sure how to define radicals generally.

Square roots

The square root of a number $x$, denoted ${\sqrt {x}}$, is the side length of a square with area $x$.

Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number $r$ such that $r^{2}=2$. It is well known that no rational number satisfies that equation. Solutions to that equation can at best be approximated by a sequence of fractions. This means that standard math's definition of ${\sqrt {2}}$ comes hand in hand with a requirement for an infinite amount of precision.

Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical square that is under consideration. [TODO I'm not completely sure about this.]

Cube roots

The cube root of a number $x$, denoted ${\sqrt[{3}]{x}}$, is the side length of a cube with volume $x$.