EXPLAIN HOW THE DEFINITION OF THE MEANING OF RATIONAL EXPONENTS FOLLOWS FROM EXTENDING THE PROPERTIES OF INTEGER EXPONENTS TO THOSE VALUES, ALLOWING FOR A NOTATION FOR RADICALS IN TERMS OF RATIONAL EXPONENTS. FOR EXAMPLE, WE DEFINE 5^(1/3) TO BE THE CUBE ROOT OF 5 BECAUSE WE WANT (5^(1/3))^3 = 5(1/3)^3 TO HOLD, SO (5^(1/3))^3 MUST EQUAL 5.
REWRITE EXPRESSIONS INVOLVING RADICALS AND RATIONAL EXPONENTS USING THE PROPERTIES OF EXPONENTS. FOR EXAMPLE: WRITE EQUIVALENT REPRESENTATIONS THAT UTILIZE BOTH POSITIVE AND NEGATIVE EXPONENTS.
EXPLAIN WHY THE SUM OR PRODUCT OF TWO RATIONAL NUMBERS IS RATIONAL; THAT THE SUM OF A RATIONAL NUMBER AND AN IRRATIONAL NUMBER IS IRRATIONAL; AND THAT THE PRODUCT OF A NONZERO RATIONAL NUMBER AND AN IRRATIONAL NUMBER IS IRRATIONAL.
Alaska: The Real Number System Math Standards