Rational Exponents and Radicals Nth Roots Principle Nth Root Examples

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

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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

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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

provided that a1/n is defined and non-zero.

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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

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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

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3 Adding and Subtracting Radicals Multiplying Radicals

Conjugates Dividing Radicals

4 Solving Equations with Exponents Odd-Root and Even-Root Property and Radicals

n Odd-Root Property

n Even-Root Property

Example