6-2 Rational Exponents
Warm Up
Simplify each expression.
1.
6
2.
3.
0
4
4.
1
10
5.
6.
–3
Holt McDougal Algebra 1
6-2 Rational Exponents
Objective
Evaluate and simplify expressions
containing rational exponents.
Holt McDougal Algebra 1
6-2 Rational Exponents
Recall that the radical symbol is used to
indicate roots. The index is the small number
to the left of the radical symbol that tells
which root to take. For example
represents
a cubic root. Since 23 = 222 = 8,
Holt McDougal Algebra 1
6-2 Rational Exponents
Another way to write nth roots is by using
fractional exponents. For example, for b >1,
suppose
b1 = b2k
1 = 2k
Square both sides.
Power of a Power Property
If bm = bn, then m = n.
Divide both sides by 2.
So for all b > 1,
Holt McDougal Algebra 1
6-2 Rational Exponents
Helpful Hint
When b = 0,
When b = 1,
Holt McDougal Algebra 1
6-2 Rational Exponents
Holt McDougal Algebra 1
6-2 Rational Exponents
1
Additional Example 1: Simplifying b
Simplify each expression.
A.
n
1
Use the definition of b n .
=7
B.
1
Use the definition of b n .
=2+3=5
Holt McDougal Algebra 1
6-2 Rational Exponents
Check It Out! Example 1
Simplify each expression.
a.
1
Use the definition of b n .
=3
b.
1
Use the definition of b n .
= 11 + 4
= 15
Holt McDougal Algebra 1
6-2 Rational Exponents
A fractional exponent can have a numerator
other than 1, as in the expression . You can
write the exponent as a product in two
different ways.
Power of a Power
Property
Definition of
Holt McDougal Algebra 1
6-2 Rational Exponents
Holt McDougal Algebra 1
6-2 Rational Exponents
Additional Example 2: Simplifying Expressions with
Fractional Exponents
Simplify each expression.
A.
B.
Definition of
= 243
Holt McDougal Algebra 1
= 25
6-2 Rational Exponents
Check It Out! Example 2
Simplify each expression.
a.
b.
Definition of
= (1)3
=8
Holt McDougal Algebra 1
=1
6-2 Rational Exponents
Check It Out! Example 2
Simplify each expression.
c.
Definition of
= 81
Holt McDougal Algebra 1
6-2 Rational Exponents
Additional Example 3: Application
Given a cube with surface area S, the volume V
of the cube can be found by using the formula
Find the volume of a cube with
surface area 54 m2.
Substitute 54 for s.
Simplify inside the parentheses.
Definition of
The volume of the cube is 27 m3.
Holt McDougal Algebra 1
6-2 Rational Exponents
Check It Out! Example 3
The approximate number of Calories C that an
2
animal needs each day is given by
,
where m is the animal’s mass in kilograms.
Find the number of Calories that an 81 kg
panda needs each day.
Substitute 81 for m.
Definition of
= 7227 = 1944
Holt McDougal Algebra 1
The panda needs 1944
Calories per day to
maintain health.
6-2 Rational Exponents
Remember that
always indicates a
nonnegative square root. When you simplify
variable expressions that contain , such as
the answer cannot be negative. But x may be
negative. Therefore you simplify
as |x| to
ensure the answer is nonnegative.
Holt McDougal Algebra 1
6-2 Rational Exponents
When n is even, you must simplify
to
|x|, because you do not know whether x is
positive or negative. When n is odd,
simplify
to x.
Holt McDougal Algebra 1
6-2 Rational Exponents
Helpful Hint
When you are told that all variables
represent nonnegative numbers, you do not
need to use absolute values in your answer.
Holt McDougal Algebra 1
6-2 Rational Exponents
Additional Example 4A: Properties of Exponents to
Simplify Expressions
Simplify. All variables represent nonnegative
numbers.
Definition of
•
Power of a Product
Property
Power of a Power Property
Simplify exponents.
Holt McDougal Algebra 1
6-2 Rational Exponents
Additional Example 4B: Properties of Exponents to
Simplify Expressions
Simplify. All variables represent nonnegative
numbers.
•
•
Power of a Product
Property
Simplify exponents.
Product of Powers
Property
Holt McDougal Algebra 1
6-2 Rational Exponents
Check It Out! Example 4a
Simplify. All variables represent nonnegative
numbers.
Definition of
Power of a Product
Property
Simplify exponents.
Holt McDougal Algebra 1
6-2 Rational Exponents
Check It Out! Example 4a
Simplify. All variables represent nonnegative
numbers.
Definition of
Power of a Product
Property
Simplify exponents.
Holt McDougal Algebra 1
6-2 Rational Exponents
Check It Out! Example 4b
Simplify. All variables represent nonnegative
numbers.
Power of a Product Property
and
Simplify.
= xy
Holt McDougal Algebra 1