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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.
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Example 1 Write
each of the following radicals in exponent form.
(a)
(b)
(c)
Solution
(a)
(b)
(c)
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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.
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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.
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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.
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Example 3 Simplify
each of the following.
(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]
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