We’ll open this
section with the definition of the radical. If n is a positive integer
that is greater twhere 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,han 1 and a is a real number then,
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.
Solution
We can also write the general rational exponent in terms of
follows.
We now need to talk about properties of radicals.
Properties
If n is a positive integer greater than 1 and both a
and b are positive real numbers then,
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.
1. All
exponents in the radicand must be less than the index.
2. Any
exponents in the radicand can have no factors in common with the index.
3. No fractions
appear under a radical.
4. 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 2 Simplify each
of the following.
Solution
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.
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.
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.
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.
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.
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.
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