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Ch. 5.6: Radical Expressions Product Property n ab a n n b **When multiplying two radicals, multiply the coefficients, then multiply the radicals** n Quotient Property n a a n b b **If there is a radical in the denominator, you must multiply both the numerator and denominator by a radical that will get the radical out of the denominator** Simplifying non-perfect roots: Numbers • Tree factor the number to its prime factors • See if you can make groups the size of the index • One group makes one of the number outside, anything not grouped goes back inside 1) 27 1. Tree factor 2) 3 108 3 3 3 3 3 2. Make groups the size of The index 3.Group makes one outside, Non-group items stays inside 9 12 3 33 4 3 3 3 3 3 2 2 3 3 3 4 Simplifying non-perfect roots: Variables • Divide the variable exponent by the index • Turn that number into a mixed number Exponent outside 1) x x Exponent inside Index 13 3 17 18 2) x y z 7 7 1 3 2 2 3 b a c x 1. Divide exponent by index 2. Follow picture above 13 1 =4 3 3 17 2 5 3 3 4 1 5 63 x y z 4 2 0 xy z 5 63 x y z 18 0 6 3 3 xy 2 Simplifying non-perfect roots: Numbers and Variables • Split the numbers and variables into two different radicals • Solve each individually • Write final answer with only one radical 13 1. Split into two problems 40 y 40 y13 4 10 2 25 2 2. Simplify each radical 13 1 6 2 2 y 6 y1 2 10 6 2 y 10 y 3. Combine into one radical Simplifying Quotients 1. Simplify the radicals individually in the numerator and denominator 2. If there is a radical in the denominator, follow these steps to remove it a. Tree factor numbers, divide variable exponents by the index b. Ask yourself, how many more do I need to make a group for the numbers, how many more do I need to make a group for the variables c. Place your answers in a radical, and multiply the numerator and denominator by your answers d. Simplify! 12 8 y Ex : 3 6x 3 8y 12 4 2y 3 3 6x 6x 2 y4 3 6x 3 66xx 3 66xx 2 y 4 3 36 x 2 3 666x x x 1. Simplify individually 2. Denominator with radical!!! a) How many 6’s are needed to make a group of 3? b) How many x’s are needed to make a group of 3? 3. Multiply numerator and Denominator by 3 6 6 x x 4. Simplify! 2 y 4 3 36 x 2 y 4 3 36 x 2 6x 3x Adding/Subtracting Radicals 1) Simplify each radical 2) Combine radicals that are the same by adding/subtracting the coefficients and keeping the radical Ex: 3 5 2 7 8 5 Ex: 20 12 45 48 11 5 2 7 2 2 5 2 2 3 3 3 5 4 4 3 2 5 2 3 3 5 4 3 52 3 FOIL: Just like with variables 1)Numbers outside radical multiply together 2) Numbers inside radical multiply together 1.(2 6 3 2)(3 2) 2 6 3 2 6 2 3 2 3 3 2 2 6 6 2 12 9 2 3 4 1. Foil 6 6 2 4 2. Multiply using rules above 3 9 2 3 2 6 6 2 2 3 9 2 3 2 3. Simplify the radicals