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Radicals Basic meaning a = x means x × x = a m a = x means x = a m Examples 2 is the number such that 2 2 = 2 3 4 27 = 3 because 3× 3× 3=27 p is the number such that ( p) 4 4 =p Important meaning a =a m 1 2 a =a ( a) m 1/m n =a n/m Why are these true? • Because I want to keep the rules of exponents. • Specifically aman=am+n • Let’s see how that works? What should 21/2 mean? • I can make it mean anything I want. • But I have a rule that I want to keep: aman=am+n • If I try to use this rule on 21/2, I get • 21/221/2=21/2+1/2=21=2 • So 21/2 is the number that when multilpied by itself gives me 2: • The square root of 2! Similarly Mystery Number: a 1/m Rule of Exponents: ( a So ( a But ) 1/m m ( a) m m =a 1m m m ) =a =a 1 = a (Slide 2) So let's pick a 1/m n =ma =a mn Why is this important? • By converting radicals to exponents, you only have to learn one set of rules: • Instead of learning rules for radicals and rules for exponents, just learn rules for exponents. Example ( a) ( a) 3 4 Example ( ) ( a) 3 a = (a 4 ) 1/2 3 1/4 a ( ) Example ( ) ( a) 3 a = (a 4 ) = (a ) (a ) 1/2 3 3/2 1/4 a ( ) 1/4 Example ( ) ( a) 3 a = (a 4 ) = (a ) (a ) 1/2 3/2 =a 3 1 + 2 4 3 1/4 a ( ) 1/4 Example ( ) ( a) 3 a = (a 4 ) = (a ) (a ) 1/2 3 3/2 =a =a 1/4 a ( ) 1/4 3 1 + 2 4 6 1 + 4 4 =a 7 4 Example ( ) ( a) 3 a 4 = (a ) = (a ) (a ) 1/2 3 3/2 =a =a = 1/4 a ( ) 1/4 3 1 + 2 4 6 1 + 4 4 =a ( a) 4 7 7 4 Convert to an expression using rational exponents: 5 5 3 2/3 a) 5 5/6 b) 5 3/4 c) 5 3/2 d) 5 e) None of the above Convert to an expression using rational exponents: 5´ 5 3 =5 5 1/2 1/3 2/3 1 1 + 2 3 a) 5 = 5 5/6 b) 5 3 2 + 3/4 6 6 = 5 c) 5 3/2 5/6 d) 5 =5 e) None of the above Warning! • We tried hard to keep everything compatible with the rules of exponents, but there is one problem we can’t fix: Even exponents lose information. Example Example: (-2) = 4 =2 2 Example If I try to use rules of exponents Example: (-2) = 4 =2 2 (-2) 2 = éë( -2 ) ùû 2 1/2 = (-2) = (-2) æ1ö 2ç ÷ è2ø 1 = -2 Example If I try to use rules of exponents Example: (-2) (-2) = éë( -2 ) ùû 2 1/2 2 = 4 =2 2 = (-2) Don’t match! = (-2) æ1ö 2ç ÷ è2ø 1 = -2 New (unavoidable) rule m x = x , when m is even m Or (Same thing): (x ) 1/m m = x , when m is even Simplifying square roots of numbers A trick to make your life easier Prime numbers • A prime number is a natural number > 1 that is only divisible* by 1 and itself • *Divisible means after you do the division you get a natural number. • Prime numbers: 2,3,5,7,11,13,17,19 Prime factorization • A prime number is a natural number > 1 that is only divisible* by 1 and itself • Factoring means turning something into a multiplication • A prime factorization is turning a number into a multiplication of primes. • Every number has exactly one prime factorization • Prime factorizations are completely factored Prime factorization examples • 6=2*3=2131 • 128=2*2*2*2*2*2*2=27 • 1500=2*2*3*5*5*5=223153 How to factor into primes • Learn your list of primes: 2,3,5,7,11,13,17,19 (are the important ones). • Take your number (1500), divide it by the first prime on your list as many times as you can: • 1500/2=750 • 750/2=375 • 375/2=187.5 BAD • 1500=2*2*375 • Repeat for the next largest prime, until you get 1 Prime factoring 1500 • • • • • Next prime is 3 1500=2*2*375 375/3=125 125/3=41+2/3 BAD 1500=2*2*3*125 Prime factoring 1500 • • • • • • Next prime is 5 1500=2*2*3*125 125/5=25 25/5=5 5/5=1 DONE 1500=2*2*3*5*5*5 Note • Note: With practice, you will find shortcuts. You don’t always have to start with smallest. Looking for numbers you recognize is faster • Ex: 1500=15*100=3*5*10*10=3*5*2*5*2*5 • 1500=2*2*3*5*5*5 • No matter how you do it, there is only one prime factorization to get. Simplifying Radicals 1500 = 2 * 2 * 3* 5* 5* 5 = 2 * 2 5* 5 3* 5 = 2 * 5 3* 5 = 10 15 Using Exponential Notation 1500 1500 = 2 * 2 * 3* 5* 5* 5 = 2 23153 = 2 2315152 = 2 * 2 5* 5 3* 5 = 2 2 52 3151 1 1 1 1 = 2 * 5 3* 5 =2 5 35 = 10 15 = 10 15 Simplifying Radicals 3 1500 = 2 * 2 * 3* 5* 5* 5 3 = 5* 5* 5 2 * 2 * 3 3 3 = 5 2*2*3 3 = 5 3 12 Using Exponential Notation 243 243 = 3* 3* 3* 3* 3 = 35 = 35 = 3* 3 3* 3 3 = 3134 = 3 33 = 3* 3* 3 = 32 31 =9 3 = 32 31 =9 3 =9 3 Divide the inside power method -m powers inside +1 power outside method 243 Repeated Multiplication Method Using Exponential Notation 3 243 3 243 = 3* 3* 3* 3* 3 = 3 = 3 3* 3* 3 3 3* 3 = 3 3233 = 3* 3 3* 3 = 31 3 32 = 33 9 = 33 9 3 Repeated Multiplication Method 3 5 Divide the inside power method 3 243 = 3 35 = 31 3 32 = 33 9 -m powers inside +1 power outside method Simplify and express as a single radical: 162 50 8 a) d) 204 2 2 b) 7 2 2 5 c) e) None of these 12 2 Simplify and express as a single radical: Prime Factoring 162 162 / 2 = 81 81 / 3 = 27 27 / 3 = 9 9/3=3 3/ 3 =1 162 = 2 * 3* 3* 3* 3 162 + 50 - 8 = 23 + 25 - 2 3 =3 2 1 4 2 1 2 2 +5 2 -2 1 1 2 = 9 2 +5 2 -2 2 = 12 2 C Simplify completely: 2 128c d 4 a) 64 | c | d 2 b) d) 4 e) None of these cd 8 8cd 2 2 c) 8cd 2 Simplify completely: = 128c d 2 = 2 cd 7 2 4 4 = 22 c d 1 6 =2 c d 3 =8 c d 2 2 2 4 1 2 2