nth Roots n The number b is an nth root of a if n Rational Exponents and Radicals u b = a n If n is a positive integer, then 1/n u 0 = 0 Peter Lo Ma014 © Peter Lo 2002 1 Ma014 © Peter Lo 2002 2 Principle nth root Examples: n Exponent 1/n when n is even n Evaluate each expressions: 1/2 u If n is a positive even integer and a is a positive u 4 1/n 1/4 real number, then a denotes the positive real u 16 th th n root of a and is called principle n root of 1/4 u -81 a. 1/3 u 8 n Exponent 1/n when n is odd 1/3 u (-27) u If n is a positive odd integer and a is any real 1/5 u -32 number, then a1/n denotes the real nth root of a. 1/3 u 0 Ma014 © Peter Lo 2002 3 Ma014 © Peter Lo 2002 4 1 Rational Exponents Important Rules n If m and n are positive integers, then m/n 1/n m m 1/n u a = (a ) = (a ) , provided that a1/n is defined. n If m and n are positive integers, then u provided that a1/n is defined and non-zero. Ma014 © Peter Lo 2002 5 Ma014 © Peter Lo 2002 6 Simplifying Expressions involving Using absolute value symbols with Variables exponents n When simplifying expressions involving rational n Simplify each expression. 1 exponents and variables, we must be careful to u 8 4 write equivalent expressions. (x y )4 1 9 u æ x ö 3 ç ÷ è 8 ø Ma014 © Peter Lo 2002 7 Ma014 © Peter Lo 2002 8 2 Expressions involving variables Radical Notation with rational exponents n Use the rules of exponents to simplify the following and give the answer in positive exponents. Assume that all variables represent positive real number. 2 4 1 1 a) x 3 x 3 c) (x 2 y-3) 2 1 1 - 2 æ ö 2 a ç x2 ÷ b) 1 d) 1 4 ç ÷ a ç 3 ÷ è y ø Ma014 © Peter Lo 2002 9 Ma014 © Peter Lo 2002 10 Product and Quotient Rules for Simplified Radical Form for Radicals Radicals of index n n A radical expression of index n is in simplified radical form if it has th u No perfect n power as factors of the radicand. u No fractions inside the radical u No radicals in the denominator. n Examples u Ma014 © Peter Lo 2002 11 Ma014 © Peter Lo 2002 12 3 Adding and Subtracting Radicals Multiplying Radicals Ma014 © Peter Lo 2002 13 Ma014 © Peter Lo 2002 14 Conjugates Dividing Radicals Ma014 © Peter Lo 2002 15 Ma014 © Peter Lo 2002 16 4 Solving Equations with Exponents Odd-Root and Even-Root Property and Radicals n Odd-Root Property n Even-Root Property Ma014 © Peter Lo 2002 17 Ma014 © Peter Lo 2002 18 Example Ma014 © Peter Lo 2002 19 5.