Since the days of the Pythagoreans, mathematicians have explored and studied this new kind of number.
Arabic scholars thought of these numbers as roots, meaning that these numbers were the orgin or source of the areas and volumes in much the same way as the roots of a tree are the source of its strength and nutrients and the root of a word is the origin of its spelling, structure and meaning. Scholars in the middle ages used the Latin term radix for these roots and, before symbols had been developed, simply wrote the word radix before a number to show that they were examining its root.
Eventually, in 1525, Christoff Rudolff (1499-1545) first used the symbol √ in his book Die Coss. Although no explanation remains as to why he did this, it appears likely that his symbol was a simplified way of writing the letter "r" for radix. His symbol was not universally accepted for some time, however, as many preferred using the letter "l" for the Latin latus (meaning the side of the square or cube).
For some time, if the root of a longer expression was required, parentheses were used. For example the root of 2+4 would be written √(2+4). It was the famous French mathematician René Descartes (1596-1650) who introduced the idea of drawing a line over the expression as a grouping symbol ... in his book, La Geometrie (1637). This line is called the vinculum (the same name that is used for the line in fractions).
We still use this notation today.
We use roots as inverse functions. In other words, finding a root is the inverse of finding a power ...
In each case, we speak of the "sixth root of 64," or the "fifth root of 32," etc.
Note that there are just two exceptions to this pattern. Because multiplying a number by itself is how we find the areas of squares, we refer to the second root more commonly as "the square root" and, because we find the volumes of cubes by multiplying a number by itself three times, we refer to the third root more commonly as "the cube root."
Also, because the most common roots that you will be finding are square roots, the small superscript "2" is usually omitted from the radical sign (as I have shown in the illustration above). Therefore, if you see a radical with no superscript, it means a square root!
Finally, we need to explain the terminology that we use.
The √ sign is called a radical sign. The number or expression contained within it is called the radicand. The bar across the top that acts as a grouping symbol and identifies the radicand (the expression that the radical applies to) is called the vinculum. When we speak of "a radical," we may be referring to the √ sign only, or to the entire expression (the radical sign with the radicand included).
When we write a radical, such as √4, we mean the positive root only (if there is a negative root also). Hence √4 = 2. If we wish to indicate both roots, we must use the ± sign. So x² = 4 would become x = ±√4 = ±2 since (2)(2)=4 and (-2)(-2)=4. Just remember that all radicals indicate the positive root only and that any other signs must be added.
When you learn about complex numbers, and search for the nth root(s) of numbers, you will discover that there are n different such roots of any number (apart from zero). In this case, the use of the radical indicates the principal (real) root of the radicand.