Using Properties of Radicals

1 : Properties of Rational Exponents [Algebra 2 (Y)] HCPS III: Standard 10: Patterns, Functions, and Algebra: PATTERNS AND SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations. Benchmark : Add, subtract, multiply, divide, and simplify rational expressions, radical expressions containing positive rational numbers, and expressions containing rational exponents. Goal: Use Properties of Radicals and rational exponents. Using Properties of Radicals Product and Quotient Properties of Radicals Property Algebra P r o d u c t P r o p e r t y ! a bn=an bn Q u o t i e n t P r o p e r t y ! abn=anbn Example 1: Use Properties of Radicals Simplify the expression. a . ) ! 163 43 b . ) ! 33 93 c . ) ! 96535 d . ) ! 48434 SIMPLIFYING Radicals A radical with index n is in s i m p l e s t f o r m if there are: No perfect nth powers in the radicand No Radicals in any denominators Example 2: Write Radicals in Simplest Form Simplify the expression.

2 A . ) ! 1043 b . ) ! 403 c . ) ! 1323 d . ) ! 184 Properties of Rational Exponents Property Algebra Example Product of Powers ! am an=am+n ! 312 332=312+32()=32=9 Power of a Power ! am()n=am n ! 232()2=232 2()=23=8 Power of a Product ! ab()m=ambm ! 9 4()12=912 412=3 2=6 Quotient of Powers ! aman=am"n ! 532512=532"12()=51=5 Power of a Quotient ! ab" # $ % & ' m=ambm ! 827" # $ % & ' 13=8132713=23 Negative exponent ! a"m=1am ! 1a"n=an ! 16"12=11612=14 Example 3: Use Properties of Rational Exponents Simplify the Expression. a . ) ! 734 714 b . ) ! 614()4 c . ) ! 49 16()12 d . ) ! 6413 e . ) ! 832812 f . ) ! 752712 Like Radicals Like Radicals have the S A M E I N DE X a n d S A M E R A D I C A N D . e . g . , ! 23 a n d ! 423 Example 4: Add or Subtract Like Radicals Simplify the expression.

3 A . ) ! 7125"125 b . ) ! 435+35 c . ) ! 4923()+8923() d .) ! 71113()"101113() Simplifying Variable Expressions Example 5: Simplify Expressions with Variables Simplify the expression. Write your answer Using positive exponents only. Assume all variables are positive. a . ) ! 16x4 b . ) ! 9x6 c. ) ! (16y8)14 d. ) ! (4y6)12 e. ) ! x6y93 f. ) ! x3y63 g. ) ! 4y54zyz"3 h. ) ! 3a32cac"2 Example 6: Add and Subtract Expressions with Variables Simplify the expression. Assume all variables are positive. a . ) ! 5y"2y b . ) ! 4x"3x c. ) ! 6x2y34+3x2y34 d. ) ! 5xy14+2xy14