Transcript Holt McDougal Algebra 1 6-2
Lesson Quiz
6-2 Rational Exponents Warm Up Simplify each expression.
1. 2. 3.
4. 5.
6. 6 0 4 1 –3 10 Holt McDougal Algebra 1
6-2 Rational Exponents
Objective
Evaluate and simplify expressions containing rational exponents.
Holt McDougal Algebra 1
6-2 Rational Exponents
Vocabulary
index
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 2 3 = 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
b
1 = b 2k 1 = 2k
Square both sides.
Power of a Power Property If b m = b n , 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 Additional Example 1: Simplifying b Simplify each expression.
1 n A.
b 1 n
= 7 B.
b 1 n
= 2 + 3 = 5 Holt McDougal Algebra 1
6-2 Rational Exponents Check It Out!
Example 1 Simplify each expression.
a.
= 3 b.
Holt McDougal Algebra 1 = 11 + 4 = 15
b 1 n 1 n
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 = 25 Holt McDougal Algebra 1
6-2 Rational Exponents Check It Out!
Example 2 Simplify each expression.
a.
b.
Definition of
= 8 = (1) 3 = 1 Holt McDougal Algebra 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 m 2 .
Substitute 54 for s.
Simplify inside the parentheses.
Definition of
The volume of the cube is 27 m 3 . 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 ’ panda needs each day. s mass in kilograms. Find the number of Calories that an 81 kg
Substitute 81 for m.
= 72
27 = 1944
Definition of
The panda needs 1944 Calories per day to maintain health.
Holt McDougal Algebra 1
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 numbers.
Simplify Expressions Simplify. All variables represent nonnegative Holt McDougal Algebra 1 • •
Power of a Product Property Simplify exponents.
Product of Powers Property
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
6-2 Rational Exponents Lesson Quiz: Part I Simplify each expression.
1. 2 2. 9 3. 729 4. 128 Holt McDougal Algebra 1
6-2 Rational Exponents 5.
Lesson Quiz: Part II In an experiment, the approximate population P of a bacteria colony is given by , where t is the number of days since start of the experiment. Find the population of the colony on the 8th day.
480 Simplify. All variables represent nonnegative numbers.
6.
7.
Holt McDougal Algebra 1