7-5 Fractional Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California...
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Transcript of 7-5 Fractional Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California...
7-5 Fractional Exponents
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.
California Standards
7-5 Fractional 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 = 222 = 8,
7-5 Fractional Exponents
Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose
1 = 2k
So for all b > 1,
Square both sides.
Power of a Power Property
If bm = bn, then m = n.
Divide both sides by 2.
b1 = b2k
7-5 Fractional ExponentsAdditional Example 1: Simplifying b
1n
Simplify each expression.
A.
= 7
b1nUse the definition of .
B.
b1nUse the definition of .
= 2 + 3 = 5
7-5 Fractional ExponentsCheck It Out! Example 1
Simplify each expression.
a.
= 3
b.
= 11 + 4
= 15
b1nUse the definition of .
b1nUse the definition of .
7-5 Fractional 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 PowerProperty
Definition of
7-5 Fractional ExponentsAdditional Example 2: Simplifying Expressions with
Fractional Exponents
Simplify each expression.
A. B.
Definition of
= 243 = 25
7-5 Fractional ExponentsCheck It Out! Example 2
Simplify each expression.
a.
= 8
b.
= 1
Definition of
= (1)3
7-5 Fractional ExponentsAdditional 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.
7-5 Fractional ExponentsCheck 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.
= 7227 = 1944
The panda needs 1944 Calories per day to maintain health.
Substitute 81 for m.
Definition of
7-5 Fractional 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.
7-5 Fractional 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.
7-5 Fractional Exponents
When you are told that all variables represent nonnegative numbers, you do not need to use absolute values in your answer.
Helpful Hint
7-5 Fractional ExponentsAdditional Example 4A: Properties of Exponents to
Simplify Expressions
Simplify. All variables represent nonnegative numbers.
Power of a Product Property
Power of a Power Property
Simplify exponents.
•
Definition of
7-5 Fractional ExponentsAdditional Example 4B: Properties of Exponents to
Simplify Expressions
Simplify. All variables represent nonnegative numbers.
Power of a Product Property
Product of Powers Property
Simplify exponents.
• •
7-5 Fractional ExponentsCheck It Out! Example 4a
Simplify. All variables represent nonnegative numbers.
Power of a Product Property
Simplify exponents.
Definition of
7-5 Fractional ExponentsCheck It Out! Example 4b
Simplify. All variables represent nonnegative numbers.
Power of a Product Property and
Simplify.
= xy
7-5 Fractional Exponents
In an experiment, the approximate population P of a bacteria colony is given by
, where t is the number of days since thestart of the experiment. Find the population of the colony on the 8th day.
5.
480
Simplify. All variables represent nonnegative numbers.
6.
7.
Lesson Quiz: Part II