10/31/11 SECTION 7.1
Determine the nth roots of numbers and evaluate radical
expressions.
•  Use the rules of exponents to evaluate or simplify expressions with
rational exponents.
•  Evaluate radical functions and find the domain of radical functions.
•
New challenge
•  What is the solution to
1 10/31/11 Roots
•  Solution to the equation
Is called the nth root of a.
Roots
•  We define the principal nth root of a to be the nth root that
has the same sign as a and we write
is not real
2 10/31/11 Inverse properties of nth powers and
exponents
•  a any real number, n > 1 integer.
1.  If a has a principal nth root, then
( a)
n
2.  If n is odd, then
n
=a
n
an = a
n
an = a
3.  If n is even, then
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Exponent and roots
3 10/31/11 A different view
•  What is
•  With that in mind
A different view
•  But, that is exactly the definition of the nth root of a.
•  So raising a number to a fractional power only makes
sense when a principal root of that number exists.
4 10/31/11 Not to stop there
•  If m is a positive integer
Using the rules we already know
1
n
1
n
a ⋅ b = (ab)
a
b
1
n
1
n
1
n
⇔
1
⎛ a ⎞ n
= ⎜ ⎟
⎝ b ⎠
n
ab = n a ⋅ n b
n
⇔
n
a n a
=
b
b
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5 10/31/11 Evaluate
Rewrite with a rational exponent.
6 10/31/11 1.  True
2.  False
Find the domain of each function
7 10/31/11 1.  Not a real number
2.  -2
3.  Ummmm???
True or false?
1.  True
2.  False
8 10/31/11 SECTION 7.2
•  Use the product and quotient rules for radicals to simplify
radical expressions.
•  Use rationalization techniques to simplify radical
expressions.
Simplifying radical expressions
•  All possible nth powered factors have been removed from
each radical
•  No fractions appear under a radical.
•  No radicals appear in the denominator of a fraction.
9 10/31/11 Simplify
10 10/31/11 Simplify
11 10/31/11 Simplify
12 10/31/11 To simplify
4
3
25
…
•
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SECTION 7.3
•  Use the distributive property to add and subtract like
radicals.
•  Use radical expressions in application problem
13 10/31/11 Combine the radical expressions, if
possible.
Combine the radical expressions, if
possible.
14 10/31/11 Combine the radical expressions, if
possible.
Combine the radical expressions, if
possible.
15 10/31/11 SECTION 7.4
Use the distributive property to multiply radical expressions.
Determine the product of conjugates.
•  Simplify quotients involving radicals by rationalizing the
denominators.
•
•
Multiply & simplify
16 10/31/11 Multiply & simplify
Multiply & simplify
(
€
3
)(
x −2
3
) (
x 2 − 2 3 x +1
)(
5 −2
)
5 +2
€
17 10/31/11 Multiplying radicals
(
)(
(3
(
(
)
x +2
x −5
)(
x − y 2 x −5 y
)(
)
x −3
3
)
x +3
)
2y +10 ( 3 4 y 2 −10)
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Divide & Simplify
18 10/31/11 Divide & Simplify
Dividing radicals expressions
5
9− 6
(3 x − y )
(2 x − 5 y )
u+v
u −v − u
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19