Algebra B: Chapter 6 Notes 1 Chapter 6: Radical Functions and Rational Eponents Concept Bte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Propert of Eponents: Product of Powers m n = m + n ( 7 )( 6 ) r (-1r ) 6cd (c d ) Propert of Eponents: Power of a Power ( m ) n = mn ( ) 9 [( ) ] (k ) Propert of Eponents: Power of a Product () m = m m ( ) (8g h ) (g h ) (9h ) Dividing Monomials: Propert of Eponents: Quotient Rule m n mn 9 8 9 c c
Algebra B: Chapter 6 Notes Zero Propert of Eponents 0 1 0 0 7 t s t 0 Propert of Negative Eponents 1 m m 8 7 0 Propert of Eponents: Power of a Quotient m m m 1 z n n b
Algebra B: Chapter 6 Notes Eample: Simplif and rewrite each epression using onl positive eponents. 1. ( a )( a ). ( ). ab c a bc 6. a b b 1.
Algebra B: Chapter 6 Notes 6.1 Roots and Radical Epressions 9 We sa is the root of 9 and write this: 6 We sa is the root of 6 and write this: 16 We sa is the root of 16 and write this: When we talk about roots: odd roots a, a, etc. are positive is a is positive and negative if a is negative. Eample: 1000 8 6 even roots a, a, a, etc. are onl possible REAL NUMBERS if a is positive. We will also onl consider the positive root/ also called the principle root. Eample: 100 6 9
Algebra B: Chapter 6 Notes Eample 1: Eample : Find the real cube roots of each number. Find the real fourth roots of each number. 0.008 16-1000 -0.0001 16 81 Eample : Eample : Find the real fifth roots of each number. Find the real square roots of each number. 0.01-1 - 6 11 Eample : Find each real number root. 1 81 7 d) 7 Radical Epressions Or, what happens when variables get involved. ** Note ** in our book, the authors take a lot of time tring to help ou understand that when variables are involved, ou need to be especiall careful about the ideas of positive/negative. The authors reall want ou to use absolute value signs to force a variable to be positive. For the purposes of our class, we will be writing our answers without these absolute value signs.
Algebra B: Chapter 6 Notes 6 Recall from Rules of Eponents B this same logic: Simplif each radical epression. 6 ( a ab 6 Eample 6: Simplif each radical epression 8 16 6 9 ab 8 1 d) 81 1 1 1a b e) 1 16 8 z Eample 7: You can use the epression D = 1. h to approimate the visibilit range D,in miles, from a height of h feet above ground. How far above ground is an observer whose visibilit range is 8 miles?
Algebra B: Chapter 6 Notes 7 6. Multipling and Dividing Radical Epressions We ve talked a little about simplifing numerical epressions as we have solved quadratic equations using the quadratic formula and b square roots. We will use these same ideas to multipl radical epressions. Recall: If a and b are positive, then ab a b. Write in simplest radical form: 00 7 8 Eample 1: Can ou simplif the product of the rational epression? If the radicand has a perfect root among its factors, ou can used the product rule to simplif. This is called simplest radical form and is NOT A CALCULATOR ESTIMATE. Eample : Write in simplest radical form. 0 160 16 d) 6
Algebra B: Chapter 6 Notes 8 We can still use our division idea to deal with variable eponents, but now let s consider epressions that don t divide evenl. Yesterda: 0 a Toda: 1 a Think: 1 Eample : Write in simplest radical form. 8 ab 7 9 z 9 Simplifing a Product: Step 1: Use the product rule to combine like radicals Step : Simplif using perfect Nth factors. Eample : What is the simplest form of the epression? d)
Algebra B: Chapter 6 Notes 9 e) 9 7 f) 8 g) h) 7 6 Quotients: Dividing Radicals Eample : Simplif the following quotients 18 16 0 6
Algebra B: Chapter 6 Notes 10 For a radical epression to be simplified No perfect square factors under radicals No radicals in denominators No denominators under radicals A frequent simplification issue: 1 8 To solve this simplification problem we are going to RATIONALIZE THE DENOMINATOR! Rationalize the denominator: Multipl the fraction b something equivalent to 1. (The same value to the top and bottom ) Goal: Create a perfect square/ perfect nth factor on the denominator. Eample 6: Rationalize the denominator of each epression: 6 9 18 d) 7 e) 8 f) a bc
Algebra B: Chapter 6 Notes 11 6. Binomial Radical Epressions Like Radicals are radicals with the same inde and the same radicands. You can multipl and divide an radicals with the same inde. HOWEVER, You can onl add and subtract LIKE RADICALS. Be especiall cautious when combining/adding radicals. 1.7 1.7.6 1.11.7.1 Eample: (1.7).6. Eample 1: What is the simplified form of each epression? 1 7 87 6 7 d) 7 e) f) 17 1 Sometimes ou ma have like radicals, but ou can t see them until ou simplif. Eample : What is the simplest form of the radical epression? 1 7 0 16
Algebra B: Chapter 6 Notes 1 Sometimes ou have to use FOIL to simplif a radical epression. Eample : What is the product of each radical epression? 7 7 6 1 6 1 d) 8 8 Notice that in parts ( and (d) that ou are multipling CONJUGATES: a b and a b An time ou multiple radical conjugates, the result is a rational number. Eample: Here our denominator is: so we want to multipl b its conjugate.
Algebra B: Chapter 6 Notes 1 Eample : Write the epression with a rationalized denominator. 7 6 6. Rational Eponents Check the following in our calculator: 1 9 1 11 1 16 1 1 1 81 1
Algebra B: Chapter 6 Notes 1 m n m n a a Dealing with Rational Eponents: 1. Rewrite the epression as a radical.. Multipl if necessar using radical rules.. Simplif if ou can. Eample 1: 1 6 1 1 11 11 1 1 1 Eample : Rewrite in simplest radical form 7 0. w 8 d). Eample : Rewrite in eponential form. a b d)
Algebra B: Chapter 6 Notes 1 Eample : What is each product or quotient in simplest form. 8 d) 7 7
Algebra B: Chapter 6 Notes 16 Eample : What is each number in simplest radical form?. 9 81 d) 16. Eample 6: Simplif each epression. 1 1 (8 )
Algebra B: Chapter 6 Notes 17 6. Solving Radical Equations An time ou have a variable under the radical sign, ou ma have to use eponents to solve. Steps Solve: 8 1. Isolate the radical. Raise each side of the equation to the nth powers. Solve the equation. CHECK YOUR ANSWERS!!!! Eample 1: 1 0 1 ( 1) 1 d) ( 1) 1
Algebra B: Chapter 6 Notes 18 e) 1 7 0 f) 10 6.8 Graphing Radical Functions Graph the following radical epression
Algebra B: Chapter 6 Notes 19 1