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Algebra 1
7.5 Fractional Exponents
California
Standards
2.0 Students understand and use such
operations as taking the opposite, finding the
reciprocal, taking a root, and raising to a
fractional power. They understand and use the
rules of exponents.
Vocabulary
index
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,
Another way to write nth roots is by using fractional
exponents. For example, for b >1, suppose
Square both sides.
b1 = b2k
1 = 2k
Power of a Power Property
If bm = bn, then m = n.
Divide both sides by 2.
So for all b > 1,
Helpful Hint
When b = 0,
When b = 1,
1
Additional Example 1: Simplifying b n
Simplify each expression.
A.
1
Use the definition of b n.
=7
B.
1
Use the definition of b n .
=2+3=5
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
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
Additional Example 2: Simplifying Expressions with
Fractional Exponents
Simplify each expression.
A.
B.
Definition of
= 243
= 25
Check It Out! Example 2
Simplify each expression.
a.
b.
Definition of
= (1)3
=8
=1
Check It Out! Example 2
Simplify each expression.
c.
Definition of
= 81
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.
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
The panda needs 1944
Calories per day to
maintain health.
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.
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.
Helpful Hint
When you are told that all variables represent
nonnegative numbers, you do not need to use
absolute values in your answer.
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.
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
Check It Out! Example 4a
Simplify. All variables represent nonnegative
numbers.
Definition of
Power of a Product Property
Simplify exponents.
Check It Out! Example 4b
Simplify. All variables represent nonnegative
numbers.
Power of a Product Property and
Simplify.
= xy
TOD Lesson Quiz 7.5
Simplify each expression.
Simplify. All variables
2
1.
represent nonnegative
numbers.
9
2.
6.
729
3.
4.
7.
128
In an experiment, the approximate population P
5.
of a bacteria colony is given by
, where t is the number of days since the
start of the experiment. Find the population of the
colony on the 8th day.
480