Algebraic Expressions Containing Radicals
A radical or root is sort of like the reverse of a power.
We’ve already covered methods for working with radicals
involving just numbers in some detail in a previous section of
these notes. In this document (and four or so that follow), we
extend methods already illustrated for radicals containing only
numbers to radicals which contain symbols as well. Algebraic
radicals are still indicated by the symbol  .
The meaning of this symbol is given by the condition
That is, by  , we mean the quantity (or expression)
whose  power is equal to x.
Remarks
(i) When n = 2, the small superscript is usually omitted, so
We call  the “square root of x” because
(ii) The superscript n in is called the order of the radical or
root. The general basic algebraic properties of roots do not
depend on the order of the root, but the actual detailed symbolic
manipulations required to work with higher order roots get more
and more complicated as the order gets higher. It isn’t
difficult to extend the methods we will illustrate here to
expressions containing higher order roots if you understand the
basic principles. However, we will not take the space to do so.
(iii) You are probably aware that roots or radicals can be
written in equivalent exponential form:
and
Simplifying products or quotients of roots may be eased in
some cases if the equivalent exponential forms are used. However,
often (and always when sums or differences of radicals are
present), conversion from radical to exponential notation will
not likely be very helpful and may actually make simplifying the
expression much more difficult.
Radicals are closely related to the operation of
multiplication. This means that there are some simple properties
for multiplying or factoring radicals. Specifically
and
There are no simple rules for breaking up radicals of sums or
for combining sums of radicals! We will emphasize this over and
over again in the notes to follow, but to alert you to a very
common error, we state here that you can never do something like
or
When written specifically for square roots, what we have just
said is that
and
but
in general. Here, x and y represent any non-negative number or
valid mathematical expression that will evaluate to a
non-negative value in the case of square roots.
(v) Radicals arise in technical applications because powers
occur in technical applications. So, if you have squares of
symbols appearing in formulas, solving problems based on such a
formula will sometimes involve square roots. A very simple
example is the following. The area, A, of a square whose sides
have length s is given by 
But then, if we know the area of the square and we need to
compute the lengths of its sides, we must use the formula
Why? Because if we need to square s to get the value of A,
then s must be the quantity whose square is A – that is, s
is the square root of A.
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