1. DEFINING, REWRITING, AND EVALUATING RATIONAL EXPONENTS (1.1.1)
SEPTEMBER 1ST, 2017

2. Vocabulary Used in the Real Number System
Real Numbers: the set of all rational and irrational numbers
Rational Number: any number that can be written as a fraction , where both and are integers and ; any number that can be written as a decimal that ends or repeats
Integer: The set of whole numbers, their negatives, and zero

3. Properties of Exponents
Zero Exponent Property
Negative Exponent Property
Product of Powers Property
Quotient of Powers Property
Power of a Power Property
Power of a Product Property
Power of a Quotient Property

4. Vocabulary Used in the Exponential Number System
Exponential Equation
or Exponential Expression?
exponent/
Power
exponent/
power
base
base
Root: the inverse of a power/exponent; the root of a number is a number that, when multiplied by itself a given number of times, equals .
For example, if
then

5. So what does an exponent of 1/2 mean?
It is the same as taking the square root,
How can you prove it?
Prove that

6. Since roots are the inverses of powers, we know that
Find out what exponent can replace the cube root by solving for x.
Therefore,

7. Use the pattern you’ve discovered to write the root equivalent to each rational exponent.

8. So can you write a rule that is always true for rewriting rational exponents as roots?

9. COMBINING THIS WITH THE POWER OF A POWER PROPERTY, WE CAN REWRITEAS

10. EX.1: EVALUATE AND EXPLAIN HOW YOU ARRIVED AT YOUR ANSWER.
NOW CONSIDER HOW TO ARRIVE AT THE SAME ANSWER BY INTERPRETING AS

11. EX.2: REWRITE USING PROPERTIES OF EXPONENTS AND ROOTS, THEN EVALUATE.

12. EX.3: REWRITE EACH OF THE FOLLOWING EXPRESSIONS IN TERMS OF POWERS AND ROOTS. DETERMINE WHETHER THE RESULT WILL BE A RATIONAL OR IRRATIONAL NUMBER BY SIMPLIFYING THE EXPRESSION AS MUCH AS POSSIBLE WITHOUT USING A CALCULATOR. (YOU MAY USE A CALCULATOR TO VERIFY YOUR ANSWER.)

13. CAN YOU WRITE A GENERAL RULE FOR HOW TO REWRITE A RATIONAL EXPONENT IN TERMS OF POWERS AND ROOTS?
BIG QUESTION: WHAT IS THE RELATIONSHIP BETWEEN A POWER AND A ROOT?