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# Download Simplifying Radicals: Part I

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To simplify a radical in which the radicand contains a perfect square as a factor Example: √729 √9 ∙√81 3∙9 Perfect Square! 27 Simplifying Radicals: Part I Vocabulary and Key Concepts Index Radical symbol 2 x Radicand Read “the square root of x.” NOTE: The index 2 is usually omitted when writing square roots. Table of Perfect Squares You may find the following table of perfect squares to be helpful when you are required to simplify square roots. Complete the table below: 1 _____ 4 2 2 = _____ 9 32 = _____ 36 = _____ 49 2 7 = _____ 64 82 = _____ 121 _____ 144 2 12 = _____ 169 132 = _____ 256 _____ 289 2 17 = _____ 324 182 = _____ 12 = 62 112 = 162 = 16 42 = _____ 81 92 = _____ 196 142 = _____ 361 192 = _____ 52 = _____ 25 102 = _____ 100 152 = _____ 225 202 = _____ 400 ALERT! Check to be sure you have simplified completely: Simplifying Square Roots MENTAL MATH: Find two factors of 72, one of which is the greatest perfect square factor. Establish order, so that you don’t omit any! 1, 72 2,36 3, 24 4, 18 6, 12 72 36 g2 6 2 8, 9 9,8 (once you have a repeated factor pair, you know that you have found ALL factors!) Simplifying Square Roots: An Alternate Method 72 8g9 NOTE: If you have a perfect square (or perfect square factor) remaining under the radical symbol, you have not simplified completely. 4g2g9 2 3 2 6 2 Simplifying Square Roots KEY: L K for perfect squares or perfect square factors. 20 18 27 32 NOTE: If you have a perfect square (or perfect square factor) remaining under the radical symbol, you have not simplified completely. To multiply, then simplify square roots when possible Simplifying Radicals: Part II Product of Square Roots PRODUCT OF SQUARE ROOTS For all real numbers x ≥ 0, y ≥ 0, √x ∙√x = √x2 = x √x ∙√y = √x∙y NOTE: Squaring a number and finding the square root are inverse operations. NOTE: Squaring a number and Roots finding the square Multiplying Square with root are inverse operations. Common Radicands √3 ∙√3 = (√3)2 = __ 3 √4 ∙√4 = (√4)2 = ____ 4 __ √5 ∙√5 = (√5)2 = ____ __ 5 (2√3)2 = 4_ ∙ _3= __ 12 (3√5)2 = 9_ ∙ 4∙ _ __ 45 (2√5)2 = _5= _5= 20 __ Multiplying Square Roots with Different Radicands √2 ∙√10 = ____ = √20 √4 ∙√20 = ____ √80 = (2√3) (5√3) = ____ = 10∙3 √9∙2= 3√2 ____ ________ ____________ √4∙5 = 2√5 ____________ √16∙5 = 4√5 30 ____________ (3√2) (5√2) = ____ 6∙2 = ____________ 12 √3 ∙√6 = (3√2) (2√6) = (5√3) (√6) = √18 = 6____ ∙ √12 = ____________ 5____ ∙ √18 = ____________ To simplify an expression containing a quotient of radicals Dividing Radicals Quotient of Square Roots QUOTIENT OF SQUARE ROOTS For all real numbers x ≥ 0, y > 0 x= x y y Simplify each expression: a. b. 18 6 24 3 ALERT! When you rationalize, you are changing Rationalizing Denominators: the form of the number, but not its value. 1 2 1 2 2 g1 2 2 2 Fraction Radical in Under √ Denominator Double Check: 1. Fraction under √ ? 2. Radical in Denominator? Rationalize More Quotients of Radicals 3 4 3 6 1 12 7 9 10 5 Summary A radical expression is in simplest form when each radicand contains no factor, other than one, that is a perfect square the denominator contains no radicals and each radicand contains no fractions. Final Checks for Understanding 1. Simplify: √3 ∙√12 2. Simplify: √2 ∙√32 3. Indicate why each expressions is not in simplest radical form. a.) 5x2 b.)√8y c.) 25 √3x 5y 7 Homework Assignments: DAY 1: Simplifying Radicals WS DAY 2: Multiplying and Dividing Radicals WS