May 18th
May 18th
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.”
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