Section 4.4. Definition of nth Root

Section 4.4: Definition of nth Root Chapter 4: Polynomial and Radical Operations Radicals expressions or equivalently nth roots • The parts of a radical expression are: radical symbol, the radicand, and the index I read this expression as: The 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑥 • The nth root of a number 𝑥 is a number that when raised to the 𝑛𝑡ℎ power equals 𝑥. 𝑛 That is: √𝑥 = 𝑟 𝑖𝑓 𝑟 𝑛 = 𝑥 3 For example: √8 = 2, 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 23 = 8 4 For example: √81 = 3, 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 34 = 81 Roots of negative numbers • If n is an odd index and 𝑥 is a negative number, then: √𝑥 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 3 For example: √−64 = −4 𝑠𝑖𝑛𝑐𝑒 (−4)3 = −64 5 For example: √−32 = −2, 𝑠𝑖𝑛𝑐𝑒 (−2)5 = −32 • If n is an even index and 𝑥 is a negative number, then √𝑥 = 𝑛𝑜𝑡 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 4 For example: √−16 = 𝑛𝑜𝑛𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 (any real number raised to the 4th power will be greater than or equal to zero. For example: √−49 = 𝑛𝑜𝑛𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 (any real number squared will be greater than or equal to zero) 𝑛 𝑛 Principal square root • • The square root symbol we are used to “√ ” is called the principal square root symbol The principal square root is often called the “positive square root” as it must reduce to a nonnegative number. That is √ ≥ 0 Order of operations with radicals. If I forget to write it, assume all variables represent positive real numbers for all radical reduction problems in this chapter. • Treat radicals as exponents: PEMDAS - P(ER)MDAS For example: 8√25 𝑡ℎ𝑖𝑠 𝑚𝑒𝑎𝑛𝑠 8 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 25. 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑓𝑖𝑟𝑠𝑡 𝑡ℎ𝑒𝑛 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑡ℎ𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 8√25 = 8 ∗ 5 = 40 • For example: 7√−32 = 7(−2) = −14 5 Here is a table of common radical values that may help you learn without needing a calculator. 0 = 0, 0 2 = 0 3 − 1 = −1, (−1) 3 = −1 5 −1 = −1, ( −1)5 = −1 1 = 1, 12 = 1 3 8 = 2, 2 3 = 8 5 32 = 2, 25 = 32 4 = 2 , 22 = 4 3 − 8 = −2, (−2) 3 = −8 5 243 = 3, 35 = 243 9 = 3, 32 = 9 3 27 = 3, 33 = 27 16 = 4, 4 2 = 16 3 64 = 4, 4 3 = 64 25 = 5, 5 2 = 25 3 125 = 5, 53 = 125 6 64 = 2, 26 = 64 36 = 6, 6 2 = 36 3 216 = 6, 6 3 = 216 6 729 = 3, 36 = 729 49 = 7, 7 2 = 49 3 343 = 7, 7 3 = 343 64 = 8, 8 2 = 64 3 1000 = 10 , 10 3 = 1000 7 −1 = −1, ( −1)7 = −1 7 128 = 2, 27 = 128 81 = 9, 9 2 = 81 100 = 10, 10 2 = 100 121 = 11, 11 = 121 4 16 = 2, 2 4 = 16 4 81 = 3, 34 = 81 4 256 = 4, 4 4 = 256 4 625 = 5, 5 4 = 625 2 144 = 12, 12 = 144 2 169 = 13, 13 = 169 2 196 = 14, 14 = 196 2 225 = 15, 15 2 = 225 256 = 16, 16 2 = 256 289 = 17 , 17 2 = 289 Section 4.4: Definition of nth Root Chapter 4: Polynomial and Radical Operations SQUARE ROOTS OF VARIABLES 2 • • • The index equals 2, even though it is not written. That is: √𝑥 = √𝑥 It is proper to use absolute values when reducing the square root of a variable. √𝑎2 asks the question what positive number squared equals 𝑎2 . 𝑇ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 "𝑎" 𝑖𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 𝑤ℎ𝑒𝑛 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 𝑒𝑞𝑢𝑎𝑙𝑠 𝑎2 Unfortunately I don’t know if the variable “a” represents a positive number or not. It is technically necessary to write the “a” under an absolute value symbol when finding the square root of “a2” because the answer needs to be positive and we don’t know if “a” represents a positive or negative number. • To avoid this technicality, we often assume all variables are positive and that will allow us to find square roots of variables without needing absolute values. 𝑛𝑡ℎ root of variables • We will assume all variables represent positive real numbers so that we can avoid absolute values. The following formula will allow us to find roots with variables: 𝑛 √ 𝑥 𝑚 = 𝑥 𝑚 ⁄𝑛 3 For example: √𝑥 12 = 𝑥 12⁄3 = 𝑥 4 , 𝐼 𝑐𝑎𝑛 𝑐ℎ𝑒𝑐𝑘 𝑚𝑦 𝑎𝑛𝑤𝑒𝑟 𝑙𝑖𝑘𝑒 𝑡ℎ𝑖𝑠 (𝑥 4 )3 = 𝑥 4∗3 = 𝑥 12 ✓ 2 For example: √𝑦 6 = √𝑦 6 = 𝑦 6⁄2 = 𝑦 3 , 𝐼 𝑐𝑎𝑛 𝑐ℎ𝑒𝑐𝑘 𝑚𝑦 𝑎𝑛𝑤𝑒𝑟 𝑙𝑖𝑘𝑒 𝑡ℎ𝑖𝑠 ∶ (𝑦 3 )2 = 𝑦 6✓ 𝑛𝑡ℎ root of products • 𝑛 𝑛 𝑛 √𝑎𝑏 = √𝑎 √𝑏 (𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑒𝑎𝑐ℎ 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟) In the following examples, assume all variables represent positive real numbers. For example: √81𝑥 8 = √81√𝑥 8 = 9𝑥 8⁄2 = 9𝑥 4 , 𝐼 𝑐𝑎𝑛 𝑐ℎ𝑒𝑐𝑘: (9𝑥 4 )2 = 92 𝑥 4∗2 = 81𝑥 8 ✓ 3 • 3 3 3 For example: √125𝑥 6 = √125 √𝑥 6 = 5𝑥 6⁄3 = 5𝑥2 , 𝑐ℎ𝑒𝑐𝑘: (5𝑥2 ) = 53 𝑥 2∗3 = 125𝑥 6✓ This rule can be used in reverse to simplify multiplication problems with radicals that have the same index 𝑛 𝑛 𝑛 √𝑎 √𝑏 = √𝑎𝑏(𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑒𝑎𝑐ℎ 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟) For example: (√4𝑥 3 )(√9𝑥) = √4𝑥 3 ∗ 9𝑥 = √36𝑥 4 =√36√𝑥 4 = 6𝑥2 𝑛𝑡ℎ root of quotients • 𝑛 𝑎 𝑛 √𝑏 = √𝑎 √𝑏 𝑛 (𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑒𝑎𝑐ℎ 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟) For example: • √ 𝑥2 100 = √𝑥 2 √100 = 𝑥 2⁄2 10 = 𝑥 2 𝑥 𝑥2 𝑥2 , 𝑐ℎ𝑒𝑐𝑘 ( ) = 2 = ✓ 10 10 10 100 This rule can be used in reverse to simplify division problems with radicals that have the same index 𝑛 √𝑎 √𝑏 𝑛 𝑛 𝑎 = √ (𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑒𝑎𝑐ℎ 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏 3 For example: √72𝑥 7 3 72𝑥 7 3 √ 9𝑥 =√ 9𝑥 3 3 3 = √8𝑥 6 = √8 √𝑥 6 = 2𝑥 2