Consider
Using the properties of exponents we would get
So the nth power of fina 1
is a .
From this we get the following definition.
Definition:
If 
and n is positive,
then 
is called the nth root of . The value
of 
is a number such that the nth power of
the number gives you a.
Example 1:
Evaluate the following.
Solution:
a. We want a number that when we raise it to the second power we get 49. Since
we have
b. Here we want a number that when raised to the third
power will give us 8. Since 
we have
c. This time we need a number that when it is raised to
the second power we get a –4. There is no such number since anything to the
second power would be a positive. Therefore, we say
is not a real number .
d. Lastly, we want a number that we can raise to the third
power and get –27. Since we have an odd power here it is possible to get a
negative. Therefore, since 
, we have
Now since we are dealing with fractional exponents, we
should review some of our basic properties and rules of exponents .
As it turns out, all of the properties and rules that we
had for exp onents before will still work when the exponent is a rational number .
Rules for Rational Exponents
Let us illustrate these rules. Remember that simplifying
an expression with exponents means leaving the ex pression with no negative
exponents and also making sure all the values of the same base have been
combined.
Example 2:
Simplify. Leave no negative exponents.
Solution:
a. According to property 1 we can write
Note, we choose to do the rational exponent inside because
it generally will make the values easier to work with. Either way is correct
however and would result in an answer of 8.
b. By property number 2, we should add the exponents in
this situation. So we get
Recall, to add ( or subtract ) fractions you need to add (or
subtract) the numerators over the LCD . So we get
c. According to property 3, we need to multiply the
exponents here. So this gives us
d. We use the second part of property 4 here. We can pull
the exponent through. When we do this always remember to multiply the value you
are pulling though by the exponents that are already there. This gives us
e. For this problem we need to get rid of the negative on
the exponent. In order to do that, we need to use property 5. The easiest way to
remember how to get rid of a negative exponent is just move the value across the
fraction bar. Anytime we do that it changes the value from a positive to a
negative exponent. So if we have a negative exponent on the bottom we move it to
the top and if its on top (as in our problem) we move it to the bottom. We get
Notice, that the only thing that changes with the exponent
is the sign. Nothing else about it changes. It is still a ½.
f. The first thing we want to do in this problem is get
the negative exponent to a positive one. So we can use property 6 to do that. We
simply “flip” the fraction inside and that will change the sign of the exponent .
So we will have
Now we use properties 4 and 1 to completely simplify the
expression. We proceed as follows
g. Now we need to use several properties together to
simplify the expression. We will begin with pulling the
through the numerator. Then we will combine
all the values into one by using both parts of property 2 and get rid of any
negative exponents we are left with. We get
h. Lastly, we will need to combine several properties
again. We start getting rid of the negative on the
by flipping the fraction inside.
Next we will pull the exponent though and get rid of the
remaining negative exponent. We will then finish any remaining simplifying that
needs to be done. We have.
Now that we have a familiarity with the rational exponents
we want to see a slightly different but more widely used notation for nth roots.
Definition:
If a is a real number, then
, n is called the index,
is called the radical symbol and the
expression underneath the radical is called the radicand.
This radical notation is going to be used frequently
throughout the rest of this text. Also, rule number 1 above gives us the
following very useful rule.
Note: If there is no index on the radical it is assumed to
be a two, we call it a square root
Rule for Radicals:
So we can see from this that we can easily change between
radical and exponent notation. We simply need to remember that the index is the
denominator and vice versa.
Example 3:
Rewrite in the alternate notation.
Solution:
a. By the above definition, we know that the denominator
becomes the index. However, when the index is 2, we need not write it.
Therefore,
b. On this example we need to be careful. Remember that
the exponent only goes with the object right before it. In this case that means
that the 
only goes with the . So the –3 will
remain out in front of the radical expression. Also, 5 will be the index, since
it is the denominator. So we have 
c. This time we are trying to go back to the exponent
notation. Notice that the entire expression 
is under the radical symbol. That means that the exponent will go to the entire
expression 
. Since the index becomes the
denominator we have
d. Lastly, we notice that the
is the only part under the radical.
Therefore, the 2y will not have the rational exponent on it. So since we see no
index, we know it is a 2. Therefore we get 
8.1 Exercises
Evaluate the following.
Simplify. Leave no negative exponents.
Rewrite in the alternate notation.
