We’ll open this section with the definition of the
radical. If n is a positive integer that is greater than 1 and a is a real number then,
where n is called
the index, a is called the radicand,
and the symbol is called the radical. The left side of
this equation is often called the radical form and the right side is often
called the exponent form.
From this definition we can see that a radical is simply
another notation for the first rational exponent that we looked at in the rational exponents section.
Note as well that the index is required in these to make
sure that we correctly evaluate the radical.
There is one exception to this rule and that is square root. For square roots we have,
In other words, for square roots we typically drop the
index.
Let’s do a couple of examples to familiarize us with this
new notation.
Example 1 Write
each of the following radicals in exponent form.
(a)
(b)
(c)
Solution
(a)
(b)
(c)

As seen in the last two parts of this example we need to be careful
with parenthesis. When we convert to
exponent form and the radicand consists of more than one term then we need to
enclose the whole radicand in parenthesis as we did with these two parts. To see why this is consider the following,
From our discussion of exponents in the previous sections we
know that only the term immediately to the left of the exponent actually gets
the exponent. Therefore, the radical
form of this is,
So, we once again see that parenthesis are very important in
this class. Be careful with them.
Since we know how to evaluate rational exponents we also
know how to evaluate radicals as the following set of examples shows.
Example 2 Evaluate
each of the following.
(a) and [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
To evaluate these we will first convert them to exponent
form and then evaluate that since we already know how to do that.
(a) These are together to make a point about the importance of
the index in this notation. Let’s take
a look at both of these.
So, the index is important. Different indexes will give different
evaluations so make sure that you don’t drop the index unless it is a 2 (and
hence we’re using square roots).
(b)
(c)
(d)
(e)
As we saw in the integer exponent section this does not
have a real answer and so we can’t evaluate the radical of a negative number
if the index is even. Note however
that we can evaluate the radical of a negative number if the index is odd as
the previous part shows.

Let’s briefly discuss the answer to the first part in the
above example. In this part we made the
claim that because . However, 4 isn’t the only number that we can
square to get 16. We also have . So, why didn’t we use 4 instead? There is a general rule about evaluating
square roots (or more generally radicals with even indexes). When evaluating square roots we ALWAYS take
the positive answer. If we want the
negative answer we will do the following.
This may not seem to be all that important, but in later
topics this can be very important.
Following this convention means that we will always get predictable
values when evaluating roots.
Note that we don’t have a similar rule for radicals with odd
indexes such as the cube root in part (d) above. This is because there will never be more than
one possible answer for a radical with an odd index.
We can also write the general rational exponent in terms of
radicals as follows.
We now need to talk about some properties of radicals.
Properties
Note that on occasion we can allow a or b to be negative and
still have these properties work. When
we run across those situations we will acknowledge them. However, for the remainder of this section we
will assume that a and b must be positive.
Also note that while we can “break up” products and
quotients under a radical we can’t do the same thing for sums or
differences. In other words,
If you aren’t sure that you believe this consider the
following quick number example.
If we “break up” the root into the sum of the two pieces we
clearly get different answers! So, be
careful to not make this very common mistake!
We are going to be simplifying
radicals shortly so we should next define simplified
radical form. A radical is said to
be in simplified radical form (or just simplified form) if each of the
following are true.
 All
exponents in the radicand must be less than the index.
 Any
exponents in the radicand can have no factors in common with the index.
 No
fractions appear under a radical.
 No
radicals appear in the denominator of a fraction.
In our first set of simplification examples we will only
look at the first two. We will need to
do a little more work before we can deal with the last two.
Example 3 Simplify
each of the following. Assume that x, y, and z are positive.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
Solution
(a)
In this case the exponent (7) is larger than the index (2)
and so the first rule for simplification is violated. To fix this we will use the first and
second properties of radicals above.
So, let’s note that we can write the radicand as follows.
So, we’ve got the radicand written as a perfect square
times a term whose exponent is smaller than the index. The radical then becomes,
Now use the second property of radicals to break up the
radical and then use the first property of radicals on the first term.
This now satisfies the rules for simplification and so we
are done.
Before moving on let’s briefly discuss how we figured out
how to break up the exponent as we did.
To do this we noted that the index was 2. We then determined the largest multiple of
2 that is less than 7, the exponent on the radicand. This is 6.
Next, we noticed that 7=6+1.
Finally, remembering several rules of exponents we can
rewrite the radicand as,
In the remaining examples we will typically jump straight
to the final form of this and leave the details to you to check.
[Return to Problems]
(b)
This radical violates the second simplification rule since
both the index and the exponent have a common factor of 3. To fix this all we need to do is convert
the radical to exponent form do some simplification and then convert back to
radical form.
[Return to Problems]
(c)
Now that we’ve got a couple of basic problems out of the
way let’s work some harder ones.
Although, with that said, this one is really nothing more than an
extension of the first example.
There is more than one term here but everything works in
exactly the same fashion. We will
break the radicand up into perfect squares times terms whose exponents are
less than 2 (i.e. 1).
Don’t forget to look for perfect squares in the number as
well.
Now, go back to the radical and then use the second and
first property of radicals as we did in the first example.
Note that we used the fact that the second property can be
expanded out to as many terms as we have in the product under the
radical. Also, don’t get excited that
there are no x’s under the radical
in the final answer. This will happen
on occasion.
[Return to Problems]
(d)
This one is similar to the previous part except the index
is now a 4. So, instead of get perfect
squares we want powers of 4. This time
we will combine the work in the previous part into one step.
[Return to Problems]
(e)
Again this one is similar to the previous two parts.
In this case don’t get excited about the fact that all the
y’s stayed under the radical. That will happen on occasion.
[Return to Problems]
(f)
This last part seems a little tricky. Individually both of the radicals are in
simplified form. However, there is
often an unspoken rule for simplification.
The unspoken rule is that we should have as few radicals in the
problem as possible. In this case that
means that we can use the second property of radicals to combine the two
radicals into one radical and then we’ll see if there is any simplification
that needs to be done.
Now that it’s in this form we can do some simplification.
[Return to Problems]

Before moving into a set of examples illustrating the last
two simplification rules we need to talk briefly about
adding/subtracting/multiplying radicals.
Performing these operations with radicals is much the same as performing
these operations with polynomials. If
you don’t remember how to add/subtract/multiply polynomials we will give a
quick reminder here and then give a more in depth set of examples the next
section.
Recall that to add/subtract terms with x in them all we need to do is add/subtract the coefficients of the
x.
For example,
Adding/subtracting radicals works in exactly the same
manner. For instance,
We’ve already seen some multiplication of radicals in the
last part of the previous example. If we
are looking at the product of two radicals with the same index then all we need
to do is use the second property of radicals to combine them then
simplify. What we need to look at now
are problems like the following set of examples.
Example 4 Multiply
each of the following. Assume that x is positive.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
In all of these problems all we need to do is recall how
to FOIL binomials. Recall,
With radicals we multiply in exactly the same manner. The main difference is that on occasion
we’ll need to do some simplification after doing the multiplication
(a)
As noted above we did need to do a little simplification
on the first term after doing the multiplication.
[Return to Problems]
(b)
Don’t get excited about the fact that there are two
variables here. It works the same way!
Again, notice that we combined up the terms with two
radicals in them.
[Return to Problems]
(c)
Not much to do with this one.
Notice that, in this case, the answer has no
radicals. That will happen on occasion
so don’t get excited about it when it happens.
[Return to Problems]

The last part of the previous example really used the fact
that
If you don’t recall this formula we will look at it in a
little more detail in the next section.
Okay, we are now ready to take a look at some simplification
examples illustrating the final two rules.
Note as well that the fourth rule says that we shouldn’t have any
radicals in the denominator. To get rid
of them we will use some of the multiplication ideas that we looked at above
and the process of getting rid of the radicals in the denominator is called rationalizing the denominator. In fact, that is really what this next set of
examples is about. They are really more
examples of rationalizing the denominator rather than simplification examples.
Example 5 Rationalize
the denominator for each of the following. Assume that x is positive.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
There are really two different types of problems that
we’ll be seeing here. The first two
parts illustrate the first type of problem and the final two parts illustrate
the second type of problem. Both types
are worked differently.
(a)
In this case we are going to make use of the fact that . We need to determine what to multiply the
denominator by so that this will show up in the denominator. Once we figure this out we will multiply
the numerator and denominator by this term.
Here is the work for this part.
Remember that if we multiply the denominator by a term we
must also multiply the numerator by the same term. In this way we are really multiplying the
term by 1 (since ) and so aren’t changing its value in
any way.
[Return to Problems]
(b)
We’ll need to start this one off with first using the
third property of radicals to eliminate the fraction from underneath the
radical as is required for simplification.
Now, in order to get rid of the radical in the denominator
we need the exponent on the x to be
a 5. This means that we need to
multiply by so let’s do that.
[Return to Problems]
(c)
In this case we can’t do the same thing that we did in the
previous two parts. To do this one we
will need to instead to make use of the fact that
When the denominator consists of two terms with at least
one of the terms involving a radical we will do the following to get rid of
the radical.
So, we took the original denominator and changed the sign
on the second term and multiplied the numerator and denominator by this new
term. By doing this we were able to
eliminate the radical in the denominator when we then multiplied out.
[Return to Problems]
(d)
This one works exactly the same as the previous
example. The only difference is that
both terms in the denominator now have radicals. The process is the same however.
[Return to Problems]

Rationalizing the denominator may seem to have no real uses
and to be honest we won’t see many uses in an Algebra class. However, if you are on a track that will take
you into a Calculus class you will find that rationalizing is useful on
occasion at that level.
We will close out this section with a more general version
of the first property of radicals.
Recall that when we first wrote down the properties of radicals we
required that a be a positive
number. This was done to make the work
in this section a little easier.
However, with the first property that doesn’t necessarily need to be the
case.
Here is the property for a general a (i.e. positive or
negative)
where is the absolute value of a. If you don’t recall
absolute value we will cover that in detail in a section in the next chapter. All that you need to do is know at this point
is that absolute value always makes a
a positive number.
So, as a quick example this means that,
For square roots this is,
This will not be something we need to worry all that much
about here, but again there are topics in courses after an Algebra course for
which this is an important idea so we needed to at least acknowledge it.