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A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 A.9 DAY 2 n th Roots Rational Exponents Radical Equations Objectives: Work with n th Roots Simplify Radicals Jan 21­1:37 PM 1 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Warm‐up List the squares and cubes of the numbers 1‐10. Follow this format. 12 = 22 = 32 = 42 = 52 = 62 = 72 = 82 = 92 = 102 = 13 = 23 = 33 = 43 = 53 = 63 = 73 = 83 = 93 = 103 = Nov 3­7:32 AM 2 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Roots A radical sign is used to indicate a root. The index gives the degree of the root. The number under the radical sign is the radicand. Jan 21­2:32 PM 3 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Roots Every positive real number has two square roots, one positive and one negative. For example: The two square roots of 49 are 7 and ‐7. The positive square root is also known as the principal square root and is denoted with a radical sign: √49 = 7. Jan 21­2:32 PM 4 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Even Roots n √an = |a| if n is even and n ≥ 2. Simplify each expression. 1. √25 2. √81 3. √-100 Jan 23­1:58 PM 5 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Roots Every real number has exactly one cube root. For example: 3 √8 = 2 AND 3 ‐8 = ‐2 √ Jan 21­2:32 PM 6 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Odd Roots √an = a if n is odd and n ≥ 3. n Simplify each expression. 1. ∛27 2. ∛-8 16 3. ∛­ 125 Jan 23­1:58 PM 7 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Simplifying Radicals Jan 23­2:10 PM 8 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Example: Simplify each expression. Assume that all variables are positive. a. √72x3 b. ∛80n5 c. √50x4 d. ∛18x4 Jan 19­1:24 PM 9 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Example: Multiply. Simplify if possible. a. √2 √8 b. ∛­5 ∛25 c. √-2 √8 d. √3 √12 e. ∛3 ∛­9 f. ∜4 ∜­4 Jan 19­1:23 PM 10 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Example: Multiply and simplify. Assume that all variables are positive. a. ∛54x y ∛5x y 2 3 3 4 b. 3√7x3 2√21x3y2 ∛270x5y7 ∛27x3y6 10x2y 3xy2∛10x2y Jan 19­1:25 PM 11 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 Example: September 22, 2010 Divide and simplify. Assume that all varaibles are positive. a. ∛32 32 = 3 = ∛­4 ­4 ∛­8 = ­2 √ b. ∛162x5 162x5 = 3 = ∛54x3 = ∛27x3(2) = 3x∛2 2 2 ∛3x 3x √ Jan 6­10:52 AM 12 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Like radicals are radical expressions that have the same index and the same radicand. We can add and subtract like radicals. Add or subtract if possible. a. 5∛x - 3∛x b. 4√2 + 5√3 d. 7∜5 - 2∛5 c. 2√7 + 3√7 e. 4√xy + 5√xy Jan 24­1:11 PM 13 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Simplifying Before Adding or Subtracting a. 6√18 + 4√8 - 3√72 b. √50 + 3√32 - 5√18 Jan 24­1:18 PM 14 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 You can multiply radical expressions that are in the form of binomials by using FOIL. Simplify each expression by multiplying. b. (√2 - √3)2 a. (3 + 2√5)(2 + 4√5) Jan 24­1:53 PM 15 A.9 Day 2 nth Roots_Rational Exponents_Radical Equations 2010 September 22, 2010 Homework: page 1037 (7 ­ 16, 19 ­ 22, 25 ­ 31) Jan 23­2:10 PM 16