DEFINING, REWRITING, AND EVALUATING RATIONAL EXPONENTS (1.1.1) SEPTEMBER 1ST, 2017
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
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
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
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
Since roots are the inverses of powers, we know that Find out what exponent can replace the cube root by solving for x. Therefore,
Use the pattern you’ve discovered to write the root equivalent to each rational exponent.
So can you write a rule that is always true for rewriting rational exponents as roots?
COMBINING THIS WITH THE POWER OF A POWER PROPERTY, WE CAN REWRITE AS
EX.1: EVALUATE AND EXPLAIN HOW YOU ARRIVED AT YOUR ANSWER. NOW CONSIDER HOW TO ARRIVE AT THE SAME ANSWER BY INTERPRETING AS
EX.2: REWRITE USING PROPERTIES OF EXPONENTS AND ROOTS, THEN EVALUATE.
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.)
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?