Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m + n ( 7 )( 6 ) r (-1r ) 6cd (c d ) Property of Eponents: Power of a Power ( m ) n = mn ( ) 9 [( ) ] (k ) Property of Eponents: Power of a Product (y) m = m y m ( y) (8g h ) (g h ) (9h ) Dividing Monomials: Property of Eponents: Quotient Rule m n mn 9 y y 8 9 c c
Algebra B: Chapter 6 Notes Zero Property of Eponents 0 1 0 y 0 7 y t s t 0 Property of Negative Eponents 1 m m 8 y y 7 0 y Property of Eponents: Power of a Quotient m m m y y y 1 z n n b
Algebra B: Chapter 6 Notes Eample: Simplify and rewrite each epression using only positive eponents. 1. ( a )( a ). ( y ). ab c a bc 6. a b b 1. y 7 y (6) 0
Algebra B: Chapter 6 Notes 6.1 Roots and Radical Epressions 9 We say is the root of 9 and write this: 6 We say is the root of 6 and write this: 16 We say 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 only possible REAL NUMBERS if a is positive. We will also only 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 1 7 16 81 Eample : Eample : Find the real fifth roots of each number. Find the real square roots of each number..01-1 - 0 6 11 Eample : Find each real number root. 1 81 7 d) 7 Radical Epressions Or, what happens when variables get involved. ** Note ** in your book, the authors take a lot of time trying to help you understand that when variables are involved, you need to be especially careful about the ideas of positive/negative. The authors really want you 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 By this same logic: Simplify each radical epression. 6 ( a 8 ab y 6 y Eample 6: Simplify each radical epression 8 16 6 9 ab 8 1 y d) 81 1 1 1a b e) 1 16 8 y z Eample 7: You can use the epression D = 1. h to approimate the visibility range D, in miles, from a height of h feet above ground. How far above ground is an observer whose visibility range is 8 miles?
Algebra B: Chapter 6 Notes 7 6. Multiplying and Dividing Radical Epressions We ve talked a little about simplifying numerical epressions as we have solved quadratic equations using the quadratic formula and by square roots. We will use these same ideas to multiply radical epressions. Recall: If a and b are positive, then a b a b. Write in simplest radical form: 00 7 8 Eample 1: Can you simplify the product of the rational epression? 7 7 If the radicand has a perfect root among its factors, you can used the product rule to simplify. 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 evenly. Yesterday: 0 a Today: 1 a Think: 1 Eample : Write in simplest radical form. 8 ab 7 9 y z 9 y Simplifying a Product: Step 1: Use the product rule to combine like radicals Step : Simplify using perfect Nth factors. Eample : What is the simplest form of the epression? 6 y y y 10y d) 1 y y
Algebra B: Chapter 6 Notes 9 e) 0 f) 8 g) 0 z 1y z h) 7 y 6 y Quotients: Dividing Radicals Eample : Simplify the following quotients 18 16y y 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: Multiply the fraction by 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 y y d) 7y e) 8y 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 multiply and divide any radicals with the same inde. HOWEVER, You can only add and subtract LIKE RADICALS. Be especially cautious when combining/adding radicals. Eample: 1.7 1.7 (1.7).6.6 1.11.7..1. Eample 1: What is the simplified form of each epression? 0 0 6 7 1 7y 8 7y d) 7 1 e) y y f) 17 1 Sometimes you may have like radicals, but you can t see them until you simplify. Eample : What is the simplest form of the radical epression? 1 7 0 16
Algebra B: Chapter 6 Notes 1 Sometimes you have to use FOIL to simplify a radical epression. Eample : What is the product of each radical epression? 7 7 6 16 1 d) 8 8 Notice that in parts ( and (d) that you are multiplying CONJUGATES: a b and a b Any time you multiple radical conjugates, the result is a rational number. Eample: Here our denominator is: so we want to multiply by 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 your 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.. Multiply if necessary using radical rules.. Simplify if you can. Eample 1: 1 6 1 1 11 11 1 1 1 Eample : Rewrite in simplest radical form 7 0. 8 w d) y 7 Eample : Rewrite in eponential form. a b d) y
Algebra B: Chapter 6 Notes 1 Eample : What is each product or quotient in simplest radical form? 8 8 y y y y 1 1 1 1 y y d) 1 1 a b c
Algebra B: Chapter 6 Notes 16 Eample : What is each number in simplest radical form? 9 81 d) 16 7 Eample 6: Simplify each epression. (9y ) 1 1 (8 )