12.1 Rational Exponents
12.2 Simplifying Radicals
• A square root is the number in which we
multiply by itself 2 times to get the number
under the root symbol.
• The number under the root symbol is called
the Radicand.
•
64
this is a symbol that represents the
number that we multiply by itself 2 times to
get 64. What is that number?
• Likewise this is the symbol for the number
that we multiply by itself two times to get 15.
15
• When there is not a small number out in front
of the root symbol then it is an understood 2.
Which we call the square root.
• The number that is out front as part of the
symbol is called the INDEX.
15  15  square root
2
3
8  cube root
4
81  fourth root
5
32  fifth root
• The bottom of page 829 has a list of the most
commonly used perfect squares, cubes, and
4th roots.
• We cannot take the square root (or any even
root) of a negative number. That is because
you can never take two numbers (that are
actually the same number) that are both
negative and multiply them together and get a
negative number.
16
• However, you can take the cube root (or any
odd root) of a negative number because three
negative numbers multiplied together will
result in a negative number.
3
8
• Sometimes there will be variables that are
placed underneath the radical symbol. The
general practice is to simplify the radicals,
therefore we will try to reduce the exponents
under a radical by drawing a connection
between the INDEX of the radical and the
exponent of the RADICANDS.
2
2 4
16a b
Simplify
4 8
2
81a b
3
3
4
8x y
9
4 8
81a b
Radicals can be written using
exponents
• If x is a real number and n is a positive integer
greater than 1, then
1n
x
 x
n
1
Examples to Simplify
12
9
13
27
49
12
(49)
12
12
 16 
 
 25 
• Sometimes it is easier to write expressions using
rational exponents and use properties of exponents
(usually power to power) to simplify.
3
4
3
3
8x y
9
4 8
81a b
6
x y
12
• The rational exponent theorem allows us to
pull apart rational exponents (reverse of
power to a power rule). For example.
8
can be re  written as  (8)

13 2
23
32
25
9
2 3
x x
12
14
x 
35 56
34
z
23
z
( x1 3 y 3 )6
4 10
x y
Simplify the following
12.2 Simplifying Radicals
When simplifying radicals it is best to deal with the numerical parts
separate.
We will conduct what is called the PRIME FACTORIZATION of the
radicand.
-That means we will break the radicand down to the factors that
multiply together to give the radicand. We will then group them based
on the INDEX.
12
72
3
48
3
120
Now we can do the same thing with variables
under the radical. You will want to re-write the
variables that have higher exponents in
multiples that relate directly to the INDEX
For example…
5
2
x
3
8
y
Now put it all together inside one
problem
Simplifying Fractions under a radical
Break it up into two separate radicals. Numerator and denominator
Then simplify each radical as far as possible.
5
9
25
81
• When you do this and a radical is left in the
denominator you need to do what is called
“Rationalizing the denominator”. To do this we
multiply by a convenient value of 1. (that value will
always be whatever the denominator is)
• After you multiply you may have to reduce the
radicals again.
25
72
• So far we have assumed that all variables were
nonnegative (0 or positive). Now we want to
think about them “possibly” being negative.
• Remember though we do not know what the
variable represents so it could be positive or it
could be negative.
• This shows that the number being squared could
be 3 or it could be -3 and still get the same 9
2
• So lets look at
x we do not know
whether x represents a positive or negative
number. So when we bring it out of the
radical we are not sure whether it should be
positive or negative.
• If we are assuming that we do not know
anything about the numbers being
nonnegative then whenever a value comes
out of the radical we have to put it inside
absolute value bars.