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Warm-up • Simplify 1. 2. 3. 4. 62 (-14)2 -92 02 1. 2. 3. 4. 36 196 -81 0 Simplifying Radicals Essential Question • How do I evaluate and approximate square roots? Square root of a # • If b2 = a, then b is a square root of a. Example: if 32 = 9, then 3 is a square root of 9 Definition of Square Root • If a is a # greater than or equal to zero, the a represents the principal, or positive, square root of a and a negative sq. rt. is represented by a Examples: 9 =3 - 9 = -3 Radical and Radicand • What are they? Radical sign k Positive or Negative Radicand: # or expression under radical symbol Perfect Squares • Numbers whose square roots are integers or quotients of integers. Examples: 4, 16, 25, 100, ¼ Example 1 • Simplify 1. 2. - 25 = -5 16 = 4 3. 1 1 = 16 4 What if the radicand is not a perfect square? • You can do one of 2 things… – Give an approximation – Give a simplified exact answer Read the directions to see which one you should do! Example 1 • Simplify. Give an exact answer. 8 Write the prime factorization of 8! 2 2 2 Now circle your pairs! Pull out one number and throw out the other one. What is left? 2 2 Example 2 • Simplify. Give an exact answer. 24 Write the prime factorization of 24! 2 2 2 3 Now circle your pairs! Pull out one number and throw out the other one. What is left? 2 6 Example 3 • Simplify. Give an exact answer. 80 Write the prime factorization of 80! 2 2 2 2 5 Now circle your pairs! Pull out one number from each pair and throw out the other ones. What is left? 4 5 Example 4 • Simplify. Give an exact answer. 5 18 5 2 33 5 3 2 =15 2 Product Property • The square root of a product equals the product of the square roots of the factors. a b ab Example: 3 5 15 Example 1 (Simplify) 3 6 18 Now simplify!! 33 2 3 2 Example 2 (Simplify) a. 2 3 4 2 8 6 b. 20 12 3 4 60 48 60 2 2 2 2 3 2 2 60 3 240 3 Distribute. • Multiply/distribute . 2 6 12 6 2 24 This is simplified. Can’t add. Radicands are different. 6 2 2 2 23 6 2 2 6 Use FOIL • Multiply. (3 2 )( 4 2 ) 12 3 2 4 2 2 10 2 F irst O utside I nside L ast Always write number term before radical term! Quotient Property • The square root of a quotient equals the quotient of the square roots of the numerator and the denominator. a b a b when a and b are positive numbers Example: 9 25 9 3 25 5 Example 1 (Simplify) 3 49 3 49 3 7 Rationalizing Denominators • For example.. 3 2 2 3 It is perfectly fine to have a radical in your NUMERATOR. It is NOT o.k. to leave a radical in your DENOMINATOR! Example 2 (Simplify) 3 2 3 2 2 2 This is just a fancy form of the number 1 3 2 4 3 2 2 Example 3 (Simplify) 11 3 11 3 3 3 33 9 33 3 Example 6 150 6 25 5 You don’t have to rationalize. Just divide!! Conjugates Expression Conjugate Product a b a b a b 4 x 16 x 4 x c 2 y 7 c 2 y 7 2 c 2 2 y 7 2 Example 3 4 2 3 4 2 4 2 4 2 3( 4 2 ) 4 2 (4 2 ) 12 3 2 16 2 12 3 2 14 Example 5 • Give and approximation. Round to the nearest hundredth. 78 ≈ 8.83 Example 6 • Give and approximation. Round to the nearest hundredth. 18 ≈ 4.24 Homework • Page 144 – Numbers 1-12 all, 14, 16, 22.