Section 4.7 Answer Key

You must know your multiplication tables very well for the next two sections. This table may help a bit. 12 1*12 2*6 3*4 24 1*24 2*12 3*8 4*6 34 1*34 2*17 46 1*46 2*23 58 1*58 2*29 70 1*70 2*35 5*14 7*10 82 1*82 2*41 14 1*14 2*7 25 1*25 5*5 35 1*35 5*7 48 1*48 2*24 3*16 4*12 6*8 60 1*60 2*30 3*20 4*15 5*12 6*10 72 1*72 2*36 4*18 8*9 100 1*100 2*50 4*25 5*20 10*10 15 1*15 3*5 26 1*26 2*13 36 1*36 2*18 3*12 4*9 6*6 50 1*50 2*25 5*10 62 1*62 2*31 74 1*74 2*37 121 1*121 11*11 16 1*16 2*8 4*4 27 1*27 3*9 38 1*38 2*19 52 1*52 2*26 4*13 63 1*63 3*21 7*9 75 1*75 3*25 5*15 144 1*144 2*72 3*48 4*36 18 1*18 2*9 3*6 28 1*28 2*14 4*7 40 1*40 2*20 4*10 5*8 54 1*54 2*27 3*18 6*9 64 1*64 2*32 4*16 8*8 76 1*76 2*38 4*19 169 1*169 13*13 20 1*20 2*10 4*5 30 1*30 2*15 3*10 5*6 42 1*42 2*21 3*14 6*7 55 1*55 5*11 65 1*65 5*13 77 1*77 7*11 225 1*225 15*15 3*75 5*45 9*25 21 1*21 3*7 32 1*32 2*16 4*8 44 1*44 2*22 4*11 56 1*56 2*28 4*14 7*8 66 1*66 2*33 3*22 6*11 78 1*78 2*39 3*26 6*13 22 1*22 2*11 33 1*33 3*11 45 1*45 3*15 5*9 57 1*57 3*19 68 1*68 2*34 4*17 80 1*80 2*40 4*20 5*16 8*10 6*24 8*18 9*16 12*12 Section 4.7: Radical reduction Chapter 4: Polynomial and Radical Operations How to determine whether a 𝑛𝑡ℎ roots is reduced or not reduced A 𝑛 𝑡ℎ root is considered reduced provided no factor under the radicand has a perfect 𝑛𝑡ℎ power as a factor. • • A square root is considered reduce provided no constant or variable under the radicand has a perfect square as a factor. A cube root is considered reduce provided no constant or variable under the radicand has a perfect cube as a factor. Determine if a 𝑛𝑡ℎ root of a constant is written in reduced form: • Step 1: List all the factors of the constant. The 𝑛𝑡ℎ is reduced if no factor is a perfect 𝑛𝑡ℎ power is factor of the constant. The 𝑛𝑡ℎ is NOT reduced if a perfect 𝑛𝑡ℎ power is a factor of the constant. Determine if the radical is written in reduced form: For example: √18 To determine if √18 is reduced, list the factors of 18. √18 is reduced if none of the factors is a perfect square √18 is not reduce if one or more of the factors is a perfect square 18 = 1 × 18 18 = 2 × 9 NOTE 9 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 32 = 9 18 = 3 × 6 Solution: √18 is NOT reduced, as the perfect square 9 is a factor of 18. For example: √12 To determine if √12 is reduced, list all the factors of 12. √12 is reduced if none of the factors is a perfect square √12 is not reduced if one or more of the factors is a perfect square 12 = 1 × 12 12 = 3 × 4 NOTE 4 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 22 = 4 12 = 2 × 6 Solution: √12 is NOT reduced, as the perfect square 4 is a factor of 12. Section 4.7: Radical reduction For example: √10 3 For example: √40 Chapter 4: Polynomial and Radical Operations To determine if √10 is reduced, list the factors of 10. √10 is reduced if none of the factors is a perfect square √10 is not reduced if one or more of the factors is a perfect square 10 = 1 × 10 10 = 2 × 5 Solution: √10 is reduced if none of the factors is a perfect square 3 To determine if √40 is reduced, list the factors of 40. 3 √40 is reduced if none of the factors is a perfect cube √40 is not reduced if one or more of the factors is a perfect cube 40 = 1 × 40 40 = 2 × 20 40 = 4 × 10 40 = 5 × 8 (note 23 = 8, 𝑠𝑜 8 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑐𝑢𝑏𝑒 3 Solution: √40 is NOT reduced, as the perfect cube 8 is a factor of 40. 3 4 For example: √20 4 To determine if √20 is reduced, list the factors of 20. 4 √20 is reduced if none of the factors is a perfect fourth power 4 √20 is not reduced if one or more of the factors is a perfect fourth power 20 = 1 × 20 20 = 2 × 10 20 = 4 × 10 4 Solution: √20 is reduced since none of the factors of 20 is a perfect fourth power. Section 4.7: Radical reduction Chapter 4: Polynomial and Radical Operations Determine if a 𝑛𝑡ℎ root of a single variable is written in reduced form: • • The 𝑛𝑡ℎ root is reduced if the exponent is less than n. The 𝑛𝑡ℎ root is NOT reduced if the exponent is greater than or equal to n. Determine if the radical is written in reduced form: 3 √𝑥 2 3 √𝑥 2 is reduced if the exponent is less than 3. 3 √𝑥 2 is not reduced if the exponent is greater than or equal to 3. Solution: 3 √𝑥 2 is reduced. 4 √𝑥 5 √𝑥 2 4 √𝑥 5 is reduced if the exponent is less than 4. 4 √𝑥 5 is not reduced if the exponent is greater than or equal to 4. Solution: 4 √𝑥 5 is not reduced. √𝑥 2 is reduced if the exponent is less than 2. √𝑥 2 is not reduced if the exponent is greater than or equal to 2. 5 √𝑥 3 Solution: √𝑥 2 is not reduced. 5 √𝑥 3 is reduced if the exponent is less than 5. 5 √𝑥 3 is not reduced if the exponent is greater than or equal to 5. Solution: 5 √𝑥 3 is reduced Section 4.7: Radical reduction Chapter 4: Polynomial and Radical Operations How to reduce a radical that is not reduced There are at least two ways to teach how to reduce a radical. I will write up the method I have had the most success with. I can teach you another method in class if you like. Steps to reduce a radical: Step 1: Find the prime factorization of the constant inside the radical and factor each variable inside the radical. Step 2: Determine the index “n” of the radical. Step 3: Form groups of size “n” Step 4: Move each group of constants or variables from inside the radical to outside the radical. Step 5: Simplify the expressions both inside and outside the radical by multiplying. Reduce the radical. Step 1: find the prime factorization of 48 3 √48 Step 2: Determine the index − The index is 3. Step 3: Form groups of 3 using the prime factorization of 48 3 3 √48 = √2 × 2 × 2 × 2 × 3 Step 4: Move each group of 3 outside the radical and leave the rest under the radical. 3 3 √48 =2√2 × 3 Step 5: Simplify 3 3 Answer: √48 = 2√6 Section 4.7: Radical reduction Chapter 4: Polynomial and Radical Operations Reduce the radical. Step 1: Find the prime factorization of 𝑥 7 √𝑥 7 𝑥 7 = 𝑥𝑥𝑥𝑥𝑥𝑥𝑥 Step 2: Find the index. The index is 2. Step 3: Form groups of 2 using the prime factorization √𝑥𝑥𝑥𝑥 𝑥𝑥𝑥 Step 4: Move each group of 2 outside the radical and leave the rest under the radical. √𝑥 7 = 𝑥𝑥𝑥√𝑥 Step 5: Simplify Answer: √𝑥 7 = 𝑥 3 √𝑥 Section 4.7: Radical reduction Reduce the radical: Chapter 4: Polynomial and Radical Operations Step 1: Find the prime factorization of 300 𝑎𝑛𝑑 𝑜𝑓 𝑥 3 √300𝑥 3 300 = 2 × 2 × 3 × 3 × 5 𝑥 3 = 𝑥𝑥𝑥 Step 2: The index is 2. Step 3: Form groups of 2 using the prime factorization 300 = 2 × 2 × 3 × 3 × 5 𝑥 3 = 𝑥𝑥𝑥 √300𝑥 3 =√2 × 2 × 3 × 3 × 5𝑥𝑥𝑥 Step 4: Move each group of 2 outside the radical and leave the rest under the radical. √300𝑥 3 = 2 × 3𝑥√5𝑥 Step 5: Simplify 𝐴𝑛𝑠𝑤𝑒𝑟: √300𝑥 3 = 6𝑥√5𝑥