Radicals
The Basics
(0) In this discussion we will ONLY deal with REAL
quantities – that is, no Complex Numbers
(1) The index for any radical can only be a positive
integer greater than 1. For such indices, we make these definitions:
If n is odd,

If n is even and , x ≥ 0
where b ≥ 0 and

For odd indices can be positive, zero or negative and the
resulting will be positive, zero or negative
respectively. For example, However, for even
indices the radicand MUST be nonnegative and the resulting
will also be nonnegative. For example, but
is not defined. (Actually, it can be defined,
but it involves i, and we’ve agreed not to use i here.)
(4) We have another notation for
that uses exponents:
These fractional exponents obey all the laws
of exponents. For example, “when you multiply the bases, you add the exponents”:
However, we haven’t defined objects like yet
so we won’t push this idea here. So far we’ve only defined the meaning of
fractional exponents if the numerator is 1 (a unit fraction).
(5) On your calculator you use parentheses around the
exponent. For example, is entered on the
calculator as . Similarly,
would be entered as
(6) We have the following rule.
If n is odd,

If n is even and
,

These rules make sense if you look at the exponent form:
Thus, you can eliminate the radical if the
radicand is an object raised to a power which is a multiple of the index. For
example,
(7) Note: This is
obvious if x is nonnegative: It really
comes into play if x is negative: which is
NOT what we started with. In this case which
is what the rule requires.
Some Manipulative Skills
(1) A basic definition:
If m and n are integers with
then
provided is real.
We have the option on
to (a) root first, then power or (b) power first, then root. However, the “root
first, then power” part of the definition is the ONLY WAY we want to do these
things. It is in keeping with the idea of make small before making big that we
saw long ago. So, for example:
(2) Other manipulations:

mult. & div. only > put power throughthe product 

divide the bases > subtract the exponents 


                            
                    

mult. & div. only > flip LAST 

divide the basis > subtract the exponents 

now flip 



indices different > convert to fractional
exponents 
(3) More rules:
provided at least one
of and is real
Remember:
There is NO RULE for simplifying a radicand that is a sum
or difference:
We combine the rules above with the following rule to not
only manipulate but also simplify radicals:
(S)
provided if n is even
For example, to simplify
we begin by finding the largest multiples of
3 that do not exceed the given exponents: 3, 12, and 9 respectively in this
case. We “peel these powers off” Now, we can
apply (S) along with (M1) all at once after we note that
and