Higher Roots

Just as any positive number can be considered to be the square of other numbers (its square roots), so it is possible to express numbers as the third power, the fourth power, etc., of other numbers. So, for example,

In this case, we would say that 4 is the cube root, written

The notation stands for the number that satisfies

That is, what we mean by is the number that, when multiplied by itself n times, give the result b. So, as above,

(by definition)

and since

we conclude that .

Examples:

and so on.

The small number n in is called the order of the root (some books also call it the index of the root).

Notice that when n = 2 (square roots), this label is usually omitted, so is the same thing as .

Even order roots only exist for positive numbers, but odd order roots exist for both positive and negative numbers, as illustrated in the examples above.

Terminology

When n = 2, = is called the square root of b.

When n = 3, is called the cube root of b.

For values of n greater than 3, we just use the ordinal name for the root:

is the fourth root of b

is the fifth root of b

is the sixth root of b,

and so on.

What Are Roots Good For?

Low order roots occur commonly in technical applications. For example, the formula for the area, A, of a square with sides of length s, is

But, this means that the length of the side, s, is equal to the square root of the area, A:

A second similar example – the volume, V, of a cube which has edges of length s is .

Thus,

Because roots arise in solving important technical problems, we will explore their basic properties in the next few documents in these notes.

Radicals

The word radical is used to refer to roots or expressions involving roots. Thus , , , etc., are all called “radicals.”