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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Aim: How do I tame radicals? Simply simplify! Do Now: Find the solution set and graph the inequalities x 2 – 7x + 10 < y and x 2 – 7x + 10 < 0

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Square Root The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product that is the original number The Radical Square Root Radicand Radical sign Every positive radicand has a positive and negative sq. root. The principal Sq. Root of a number is the positive sq. root. A rational number can have a rational or irrational sq. rt. An irrational number can only have an irrational root.

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Model Problems Find to the nearest tenth: = 7.9 = 232 Find the principal Square Root: 225529 = +15= +23 = 13.4 = 11.4 = 64.4 Simplify: = |x| = 2x 8 = x + 1 = x

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Index of 2 radical sign radicand index of a number is one of the two equal factors whose product is that number Every positive real number has two square roots The principal square root of a positive number k is its positive square root,. has an index of 2 If k < 0, is an imaginary number Square Root Index of 2

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Cube Root Index = 3 Index of 3 radical sign radicand index of a number is one of the three equal factors whose product is that number has an index of 3 principal cube roots

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. n th Root The n th root of a number (where n is any counting number) is one of n equal factors whose product is that number. k is the radicand n is the index is the principal n th root of k 2 5 = 32 (-2) 5 = -32 5 4 = 625

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Index of n radical sign radicand index of a number is one of n equal factors whose product is that number principal odd roots principal even roots has an index where n is any counting number n th Root Index of n

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Radical Rules! T True or False: T T

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. simplified Radical Rule #1 In general, for non-negative numbers a, b and n Example: = x 4 Hint: will the index divide evenly into the exponent of radicand term? = x 3

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Radical Rule #2 True or False: TT If and Transitive Property of Equality If a = b, and b = c, then a = c In general, for non-negative numbers a, b, and n Example: then

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Perfect Squares – Index 2 12 144 11 121 100 10 9 9 81 8 8 64 7 7 49 6 6 36 5 5 25 4 4 16 3 3 9 4 2 2 1 1 1

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Perfect Square Factors Find as many combinations of 2 factors whose product is 75 Find as many combinations of 2 factors whose product is 128 Factors that are Perfect Squares

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Simplifying Radicals answer must be in radical form. Simplify: Find as many combinations of 2 factors whose product is 80 perfect square comes out from under the radical To simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand. The square root of the perfect square becomes a factor of the coefficient of the radical.

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Perfect Cubes 1 3 = 1 2 3 = 8 3 3 = 27 4 3 = 64 5 3 = 125 6 3 = 216 7 3 = 343 (x 4 ) 3 = x 12 (-2y 2 ) 3 = -8y 6

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Simplifying Radicals answer must be in radical form. Simplify: 2) Express the radical as the product of the roots of the factors 3) Simplify the radical containing the largest perfect power (cube) 1) Factor the radicand so that the perfect power (cube) is a factor

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Simplifying Radicals Simplify: 2) Express the radical as the quotient of two roots 1) Change the radicand to an equivalent fraction whose denominator is a perfect power. 3) Simplify the radical in the denominator

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Model Problems Simplify: KEY: Find 2 factors - one of which is the largest perfect square possible

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Aim: Simplifying Radicals Course: Adv. Alg. & Trig. Model Problems Simplify:

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