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Perfect Squares 1 64 225 4 81 256 9 16 100 121 289 25 36 49 144 169 196 400 324 4 =2 16 =4 25 =5 100 = 10 144 = 12 Simplify Using Perfect Square Method 147 Find a perfect square that goes into 147 147 49 3 147 49 147 7 3 3 Simplify 605 Find a perfect square that goes into 605 121 5 121 11 5 5 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4*2 = 2 2 20 = 4*5 = 2 5 32 = 16 * 2 = 4 2 75 = 25 * 3 = 5 3 40 = 4 *10 = 2 10 Simplifying Radicals by Tree Method. Simplify the following. 20 Step one, break the number down into a factor tree. 20 10 2 Step two: Circle any pairs. The pairs bust out of the klink. 2 5 So… 20 which is really just… 2 25 This guy gets away This guy gets caught and is never heard from again so it ends up like this… 2 The guy that got away 5 The guy left in prison This guy didn’t have a partner and so is left in prison But teacher, teacher, what happens to the other guy? He’s Dead, gone, went bye bye, won’t see him again! Another Example Simplify the following: 140 Step 1: Make a factor tree 140 10 14 2 So that leaves: Which is: 2 75 2 35 7 2 5 Cake Method Video on Cake Method Your Turn! Simplify the following: 1. 44 2 11 2. 64 8 3. 56 2 14 4. 250 5 10 LEAVE IN RADICAL FORM 18 = 288 = 75 = 24 = 72 = LEAVE IN RADICAL FORM 48 = 4 80 = 4 5 50 = 25 2 125 = 5 5 450 = 15 2 3 Adding or Subtracting Radicals To add or subtract square roots you must have like radicands (the number under the radical). Then you add or subtract the coefficients! 2 3 2 4 2 Sometimes you must simplify first: 2 18 2 3 2 4 2 Example 1: 4 8 5 32 4 4 2 5 16 2 42 2 54 2 8 2 20 2 12 2 Try These 3 5 5 5 3 75 48 4 5 5 18 3 5 5 5 2 20 3 80 DO NOW 6 5 5 20 18 7 32 2 28 7 6 63 * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals. Radical Product Property a b ab ONLY when a≥0 and b≥0 For Example: 9 16 9 16 144 12 9 16 3 4 12 Equal 5 2 5* 5 25 5 7* 7 49 7 8* 8 64 8 x* x x 7 2 8 2 x 2 2 x Multiplying Radicals 1. Multiply terms outside the radical together. 2. Multiply terms inside the radical together. 3. Simplify. 12 5 3 5 36 6 5 8 6 40 6 2 10 56 6 4 10 12 10 30 Multiplication and Radicals Simplify the expression: 7 10 4 15 7 4 10 15 28 10 15 28 150 28 25 6 28 5 6 140 6 Multiply then simplify 5 * 35 175 25 * 7 5 7 2 8 * 3 7 6 56 6 4 *14 6 * 2 14 12 14 2 5 * 4 20 20 100 20 *10 200 Multiplying Radicals You can multiply using distributive property and FOIL. 3 (7 3) 7 3 3 2 (5 8) 5 2 16 (6 2 )(6 5 2 4 2) 36 6 2 6 2 2 36 2 34 Multiply: You try. 5 (2 5 ) (2 5) 2 2 5 5 ( 2 5 )( 2 5 ) 4 2 5 2 5 5 9 4 5 (5 7 )(5 7) 25 7 18 Using the Conjugate to Simplify Conjugate Expression Conjugate (2 5 ) (2 5 ) (10 (10 2) ( 10 6 ) 2) ( 10 6 ) Product 4 5 1 100 2 98 10 36 26 The radical “goes away” every time To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator Radical Quotient Property a b a b ONLY when a≥0 and b≥0 For Example: 64 16 64 16 64 16 4 2 8 4 2 Equal 56 7 8 4*2 2 2 Fractions and Radicals Simplify the expressions: a. 5 7 10 b. There is nothing to simplify because the square root is simplified and every term in the fraction can not be divided by 10. Make sure to simplify the fraction. 4 12 2 4 4 3 2 4 2 3 2 2 2 3 2 2 3 c. 15 180 9 15 36 5 9 156 5 9 3 5 2 5 33 5 2 5 3 Dividing to Simplify Radicals No radicals in the denominator allowed Denominators must be “rationalized.” Multiply by 2 3 15 5 3 3 5 5 √ 1 in the form of √ 2 3 3 3 15 5 1 5 3 5 Using the Conjugate to Simplify Conjugate Expression Conjugate (2 5 ) (2 5 ) (10 (10 2) ( 10 6 ) 2) ( 10 6 ) Product 4 5 1 100 2 98 10 36 26 The radical “goes away” every time Dividing to Simplify Radicals conjugate Multiply by 2 5 3 5 1 in the form of 5 3 5 3 1 10 2 3 11 22 conjugate 10 2 3 25 3 5 3 11 Simplify: You try. 6 2 3 2 5 5 5 4 2 3 84 3 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 6 7 6 * 7 42 49 7 7 42 7 42 cannot be simplified, so we are finished. This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 5 10 1 * 2 2 10 2 2 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 3 12 3 * 12 3 3 3 3 36 Reduce the fraction. 3 3 6 3 6 Summary: To ADD and SUBTRACT To MULTIPLY COMBINE LIKE TERMS “Outside” NUMBERS x NUMBERS “Inside” NUMBERS x NUMBERS DISTRIBUTE and FOIL To DIVIDE “Rationalize” denominator using Use conjugate ALWAYS SIMPLIFY AT THE END IF YOU CAN 1