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Math 35
7.2 "Rational Exponents"
Objectives:
* Simplify expressions of the form a1=n and am=n :
* Simplify expressions with negative rational exponents.
* Use rules for exponents to simplify expressions.
* Simplify radical expressions.
Simplify Expressions of the Form a
De…nition:
" x1=n "
p
n
If n represents a positive integer greater than 1 and
1
n
x represents a real number,
:
Example 1: (Evaluating expressions of the form a1=n )
Evaluate.
a) 161=2
b)
27 1=3
8
c)
1
d) 111=2008
(81) 4
Example 2: (Simplifying expressions of the form a1=n )
Simplify each expression, if possible. Assume that the variables can be any real number.
a)
8x3
1
3
b)
64x12
1
6
h
i1
2 2
(y + 4)
c)
Summary of the De…nition of x1=n :
Let n be a natural number greater than 1 and let x be a real number,
1: If x > 0; then x1=n is the real number such that
:
2: If x = 0, then
:
(
and n is odd, then x1=n is the negative number such that
3: If x < 0
and n is even, then x1=n is not a real number.
Simplify Expressions of the Form
:
m
an
De…nition:
" xm=n "
p
If m and n represent positive integers (n 6= 1) and n x represents a real number,
:
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.2
m
Example 3: (Evaluating expressions of the form a n )
Evaluate.
1
a)
32
4
5
2
b) ( 64) 3
m
Example 4: (Simplifying expressions of the form a n )
Simplify each expression, if possible. Assume that the variables can be any real number.
a)
4x4
3
2
27m3 n3
b)
2=3
c)
32y 10
3
5
Simplify Expressions with Negative Rational Exponents
m
De…nition:
"x n "
If m and n are positive integers,
then
m
n
1
is in simpli…ed form, and x n is a real number,
:
Example 5: (Simplifying expressions with negative rational exponents)
Simplify each expression. Assume that x can represent any nonzero real number.
2
1
a)
b)
27x3 3
3
(36) 2
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.2
Use the Rules for Exponents to Simplify Expressions
We can use the rules for exponents to simplify many expressions with fractional exponents.
Example 6: (Using the rules for exponents to simplify expressions)
Simplify each expression. All variables represent positive real numbers. Write all answers using positive exponents only.
6
x8=3 x1=3
3
2
1
2
b) x 3 y 2
c)
a) 5 7 5 7
x2
Example 7: (Using the rules for exponents to simplify expressions)
Perform each multiplication and simplify when possible. Assume all variables represent positive real numbers. Write all
answers using positive exponents only.
4
1
3
b) x1=2 x
a) x 5 x 5 + x 5
1=2
x1=2
Simplify Radical Expressions
Using Rational Exponents to Simplify Radicals:
1: Change the radical expression into an exponential expression.
2: Simplify the rational exponents.
3: Change the exponential expression back into a radical.
Example 8: (Simplifying radical expressions)
Simplify the following expressions.
p
p
8
a)
x6
b) 9 27x6 y 3
c)
Page: 3
p
p
5
3
t
Notes by Bibiana Lopez