Radical Calculator

What this calculator does

So, you have a radical, a number under a root sign, and you want its value written out with the proper notation. This tool takes a number and a radical degree, evaluates the root, and shows the result using the radical symbol itself, to five decimal places.

There are two inputs, the number and the degree n, and the tool displays a plain root sign for a square root or a root sign with the degree attached for higher roots.

How to use it

1. Enter the number, the value going under the root sign.
2. Enter the radical degree (n).
3. Press Calculate.

The parts of a radical

A radical has named parts, and knowing them makes the notation far less mysterious. The root sign itself is called the radical, or the radical sign. The number tucked under it is the radicand, the quantity whose root you are taking. The small number written into the crook of the sign is the index, or degree, which says which root it is: a 3 for a cube root, a 4 for a fourth root, and so on. When there is no index shown at all, it means a square root, the index of 2 being understood. This tool follows that convention exactly, showing a bare root sign for degree 2 and an indexed one for anything higher.

A radical is a root in disguise

Behind the symbol, a radical is just a root, and a root is just a fractional exponent. The radical with radicand a and index n is the nth root of a, which is the same as a raised to the power 1 over n. So the whole notation is a compact way of writing "raise this number to a fractional power." The tool computes exactly that value. This is why the radical calculator and the root calculator arrive at the same numbers: they are two ways of showing one idea, one leaning on the symbol and one on the plain value.

Why the notation matters

If a radical is just a root, why bother with the symbol at all? Because notation shapes how easily you can read and manipulate an expression. The radical sign keeps the radicand and its degree neatly bundled, which makes radical expressions easy to spot and to combine in algebra. Being fluent in reading a radical, knowing at a glance what the radicand and index are, is the groundwork for simplifying radical expressions, rationalising denominators, and the rest of the algebra of roots. This tool is as much about getting comfortable with that notation as about the number it produces.

A worked example

Enter a number of 16 with a degree of 2, and the tool shows a plain root sign over 16 equalling 4, since 4 squared is 16. Enter 27 with a degree of 3, and it shows a root sign with a small 3, over 27, equalling 3. Enter 81 with a degree of 4, and it shows the fourth root of 81 equalling 3, since 3 to the fourth is 81.

Where the root sign came from

The radical sign has a surprisingly precise birthday. The symbol first appeared in print in 1525, in a book called Die Coss by the German mathematician Christoff Rudolff, who used it for square roots. It is often said to be a stylised letter r, for the Latin radix, meaning root. In 1637 René Descartes added the horizontal bar across the top, called the vinculum, that groups the radicand, and the practice of tucking the index into the crook of the sign was suggested a few years earlier by Albert Girard. The symbol you enter into this tool is the end of that long tidying-up.

Questions people ask

What are the parts of a radical called?

The root sign is the radical, the number under it is the radicand, and the small number in its crook is the index, or degree.

What does a root sign with no index mean?

A square root. The index of 2 is understood and usually left off.

Is a radical the same as a root?

Yes. The radical with radicand a and index n is the nth root of a, equal to a raised to the power 1 over n.

How does this differ from the root calculator?

They give the same value. This tool shows it using the radical symbol and its terminology, while the root calculator shows the plain number.

What if I want to simplify a radical expression?

The simplifying radicals calculator pulls perfect powers out from under the sign and combines radical expressions.

References

On radicals and the root sign. A radical denotes an nth root, equal to a fractional exponent, and the symbol has parts named the radical sign, radicand, and index.

1. Christopher Stover and Eric W. Weisstein, "Radical," from MathWorld, a Wolfram resource, on the radical and its equivalence to a fractional exponent.
2. "Earliest Uses of Symbols of Operation," MacTutor History of Mathematics, on the radical sign first appearing in Rudolff's Die Coss in 1525 and the later addition of the vinculum and index.