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P.4
Rational Exponents and Radicals
Copyright © Cengage Learning. All rights reserved.
Objectives
■ Radicals
■ Rational Exponents
■ Rationalizing the Denominator; Standard Form
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Radicals
3
Radicals
We know what 2n means whenever n is an integer. To give
meaning to a power, such as 24/5, whose exponent is a
rational number, we need to discuss radicals.
The symbol
means “the positive square root of.” Thus
Since a = b2  0, the symbol
a  0. For instance,
=3
because
makes sense only when
32 = 9 and
30
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Radicals
Square roots are special cases of nth roots. The nth root of
x is the number that, when raised to the nth power, gives x.
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Radicals
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Example 1 – Simplifying Expressions Involving nth Roots
(a)
Factor out the largest cube
Property 1:
Property 4:
(b)
Property 1:
Property 5,
Property 5:
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Example 2 – Combining Radicals
(a)
Factor out the largest squares
Property 1
Distributive property
(b) If b > 0, then
Property 1:
=
Property 5, b > 0
Distributive Property
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Example 2 – Combining Radicals
(c)
cont’d
Factor out 49
Property 1:
9
Simplify Radicals
This needs to be done on every single problem before you write an
answer. The general accepted practice is to never write 2/4 as an
answer we simplify to ½, the same applies here. If a radical can be
written in a simpler form then we want to do so. When changing to a
simpler form you should not impact the actual values represented.
When I change 2/4 to ½ my calculator for both gives 0.5 this tells me
that the simplification to ½ was right. We can use the same
argument for checking our simplified radicals. When you simplify
things you change the way it looks not the amount that it represents.
If checking yourself on questions with variables you can still use your
calculator just make sure that the values you choose for your initial
problem are the same values that you use for your check. If you let
x=2 in the initial problem you better let x=2 in your simplified version
too.
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11
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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
13
3
48
3
120
14
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
5
For example… 2
x
3
y8
15
Now put it all together inside one
problem
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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
17
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
18
19
20
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
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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.
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24
25
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27
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29
30
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Simplify Radicals
This needs to be done on every single problem before you write an
answer. The general accepted practice is to never write 2/4 as an
answer we simplify to ½, the same applies here. If a radical can be
written in a simpler form then we want to do so. When changing to a
simpler form you should not impact the actual values represented.
When I change 2/4 to ½ my calculator for both gives 0.5 this tells me
that the simplification to ½ was right. We can use the same argument
for checking our simplified radicals. When you simplify things you
change the way it looks not the amount that it represents.
If checking yourself on questions with variables you can still use your
calculator just make sure that the values you choose for your initial
problem are the same values that you use for your check. If you let x=2
in the initial problem you better let x=2 in your simplified version too.
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
y8
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 do not know whether x
• So lets look at
we
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.
x
Rational Exponents
54
Rational Exponents
To define what is meant by a rational exponent or,
equivalently, a fractional exponent such as a1/3, we need to
use radicals. To give meaning to the symbol a1/n in a way
that is consistent with the Laws of Exponents, we would
have to have
(a1/n)n = a(1/n)n = a1 = a
So by the definition of nth root,
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Rational Exponents
In general, we define rational exponents as follows.
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Example 4 – Using the Laws of Exponents with Rational Exponents
(a) a1/3a7/3 = a8/3
(b)
Law 1: aman = am +n
= a2/5 + 7/5 – 3/5
Law 1, Law 2:
= a6/5
(c) (2a3b4)3/2 = 23/2(a3)3/2(b4)3/2
=(
=2
)3a3(3/2)b4(3/2)
Law 4: (abc)n = anbncn
Law 3: (am)n = amn
a9/2b6
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Example 4 – Using the Laws of Exponents with Rational Exponents
(d)
cont’d
Laws 5, 4, and 7
Law 3
Laws 1 and 2
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Rationalizing the Denominator;
Standard Form
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Rationalizing the Denominator; Standard Form
It is often useful to eliminate the radical in a denominator by
multiplying both numerator and denominator by an
appropriate expression. This procedure is called
rationalizing the denominator.
If the denominator is of the form
, we multiply numerator
and denominator by
. In doing this we multiply the given
quantity by 1, so we do not change its value. For instance,
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Rationalizing the Denominator; Standard Form
Note that the denominator in the last fraction contains no
radical. In general, if the denominator is of the form
with m < n, then multiplying the numerator and denominator
by
will rationalize the denominator, because
(for a > 0)
A fractional expression whose denominator contains no
radicals is said to be in standard form.
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Example 6 – Rationalizing Denominators
Rationalize the denominator in each fraction.
(a)
(b)
(c)
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Example 6 – Solution
(a)
Multiply by
(b)
Multiply by
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Example 6 – Solution
(c)
cont’d
Property 2:
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Rational Exponents
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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.
this is a symbol that represents the number that
we multiply by itself 2 times to get 64. What is that
number?
64
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Likewise this is the symbol for the number that we multiply
by itself two times to get 15.
15 out in front of the root
When there is not a small number
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.
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15  2 15  square root
3
8  cube root
4
81  fourth root
5
32  fifth root
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The bottom of page 829 has a list of the most commonly
used perfect squares, cubes, and 4th roots.
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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) and
multiply them together and get a negative number
because when you multiply the same two numbers
together then they are either both positive or they are
both negative and in each instance the product would be
positive. So there is no way to multiply and get -16 if the
two factors have to be identical.
16
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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
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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
72
Simplify
2
81a 4b8
3
3
8x y
4
81a 4b8
9
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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
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Examples to Simplify
12
9
271 3
49
12
(49)
12
12
 16 
 
 25 
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Sometimes it is easier to write expressions using
rational exponents and use properties of
exponents (usually power to power) to simplify.
3
8 x3 y 9
4
81a 4b8
3
6
x y
12
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The rational exponent theorem allows us to pull apart
rational exponents (reverse of power to a power rule). For
example.
can be re  written as  (8)

13 2
23
8
32
25
9
2 3
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Simplify the following
x1 2  x1 4
x 
35 56
z3 4
z2 3
( x1 3 y 3 )6
x 4 y10
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Adding / Subtracting Radical
Expressions
Multiplication / Division of Radical
Expressions
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Two radicals are said to be “like radicals” or “like terms” if
they have the same index and the same radicand. Once
this is true you can add/subtract the coefficients out front of
the radical.
5 11  3 2  2 11
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To recognize like radicands you may have to simplify one
or more radicands.
3 2  4 8  2 18
81
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Again you want to make sure that both expressions are
simplified and rationalized.
Example 2
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Multiplying Radicals
When multiplying radicals.
The numbers on the outside get multiplied together.
Then the numbers on the insides get multiplied together.
At the end if the numbers started on the outside then their
product stays outside, and if they started on the inside the
product stays on the inside.
You usually have to simplify the radicands when done.
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Multiply the following
 2 5 3 12 
 2 15  4 5 
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Sometimes you might have to foil or
distribute the terms
3
12

8  2 5 4 3


5  2 2 4 6  5 14

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Dividing of Radicals
When division occurs with just one radical then we will
rationalize through the use of multiplying by a value of 1.
This value of 1 comes from whatever the denominator is.
If the denominator is a sum or difference of terms that
include radicals then we will use what is called a
CONJUGATE.
Conjugates are 2 groups of binomials that have the exact
same terms but the middle signs are opposite.
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Examples of Conjugates
(a  b)(a  b)
(2  r )(2  r )
( x  3)( x  3)
(4  2)(4  2)
The nice thing about conjugates is
that when you multiply (FOIL) the “O
and I” terms cancel each other out.
Which when dividing by radicals will
produce a process that eliminates all
radicals in the denominator.
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