Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4
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1 Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5. Simplif expressios of the form a.. Evaluate cube root fuctios. 7. Simplif expressios of the form a. 8. Fid eve ad odd roots. 9. Simplif expressios of the form a.. Evaluate the followig: a e. f.. For each fuctio, fid the idicated fuctio value. a. f ( x) x 5 ; f (5) g( x) x ; g( ) h( x) x ; h(9). Fid the domai for each of the followig: a. f ( x) 5x 0 g( x) 9 x 9. Simplif each expressio. a. ( ) (x ) 8 8x x x 5. For each fuctio, fid the idicated fuctio value. a. f ( x) x ; f (5) g( x) x 9 ; g(0). Simplif. a. 8a 8 e. 5 f. 5 g. (x ) h. 5 5 ( x ) i. ( ) Copright 0 Pearso Educatio, I ML-
2 Teachig Notes: If b a, the b is the square root of a. A square root of a egative umber is ot a real umber. The smbol is called a radical sig. The umber uder the radical sig is called the radica Together, the radical sig ad the radicad are called the radical expressio. A square root fuctio is defied b f ( x) x. For a real umber a, a a. The cube root of a real umber a is writte a, the a b meas b a There is a cube root fuctio defied b f ( x) x. I the radical expressio a, the umber is the idex. If is a odd umber, the the root is called a odd root. If is eve, the the root is called a eve root. A eve root of a egative umber is ot a real umber.. Aswers:. a e. 0 f.. a. f ( 5) 5 g ( ) 0. h ( 9) a. domai of f is{ x x } or [, ) domai of g is{ x x } or (, ]. a. x 9 x x + 5. a. f ( 5) g ( 0). a. a. ot a real umber f. g. x h. x + i. Copright 0 Pearso Educatio, I ML-7
3 Learig Objectives: a m. Use the defiitio of.. Use the defiitio of a. m Mii Lecture 0. Ratioal Expoets. Use the defiitio of a.. Simplif expressios with ratioal expoets. 5. Simplif radical expressios usig ratioal expoets. Use the radical otatio to rewrite each expressio, the simplif if possible. Assume that all variables represet positive umbers.. a. 8 ( ) ( x ). a. ( 7) 5 ( 9) Rewrite each expressio with a positive ratioal expoet. Simplif if possible.. a (8 x ) Rewrite each radical expressio with ratioal expoets.. a. 5 0x 5 x Use properties of ratioal expoets to simplif each expressio. Assume that all variables represet positive umbers. 5. a. ( 8 ) ( 7 b ) a x x Teachig Notes: The deomiator of a ratioal expoet is the idex of the equivalet radical. The umerator of a radical expoet is the power of which the radical is raise Remid studets that each base occurs ol oce i a simplified expressio. A simplified expressio should ot cotai a egative expoets. Remember, a base other tha zero raised to the zero power is the umber. A simplified expressio should ot have a zero expoets. Copright 0 Pearso Educatio, I ML-8
4 Aswers:. a. 8 8 x x. a. ( ) 8 ( 7) 9 ( 5) 5 9 Not A Real Number. a. 8 ( ) 8 ( 7) ( x 5 ) 5. a. 8. a. 5 ( 0x ) 9 8 ( 8x ) 7x 9 a x 9 b Copright 0 Pearso Educatio, I ML-9
5 Mii Lecture 0. Multiplig ad Simplifig Radical Expressios Learig Objectives:. Use the product rule to multipl radicals.. Use factorig ad the product rule to simplif radicals.. Multipl radicals ad the simplif.. Use the product rule to multipl. a. 7 x x x x. If f ( x) x 8x 8, express the fuctio, f, i simplified form.. Simplif. Assume all variables i a radicad represet positive real umbers ad o radicads ivolve egative quatities raised to eve powers. a. 90 8x 8 7 e. x z f. 0 5 x x z h. g x. Multipl ad simplif. Assume all variables i a radicad represet positive real umbers ad o radicads ivolve egative quatities raised to eve powers. a x x x 0x Teachig Notes: The product rule for radicals states: if a ad b are real umbers the a b ab. A umber that is the square of a iteger is a perfect square. A umber is a perfect cube if it is the cube of a iteger. A radical of idex is simplified whe its radicad has o factors other tha that are perfect th powers. For a o-egative real umber, a, a a. Perfect th powers have expoets that are divisible b. Aswers:. a. x x. f ( x) x. a. 0. a. x x 8 e. x z 00 f. x x x 5 5 x 5 x g. xz x h. x Copright 0 Pearso Educatio, I ML-70
6 Mii Lecture 0. Addig, Subtractig, ad Dividig Radical Expressios Learig Objectives:. Add ad subtract radical expressios.. Use the quotiet rule to simplif radical expressios.. Use the quotiet rule to divide radical expressios. Add or subtract. Be sure aswers are i simplified form.. a a. 0 5 x 8x 8 7. a x x Simplif usig the quotiet rule.. a. 8x x 0x 5 8x 5x Divide ad simplif if possible. 5. a. 00x x 5 0x 5 5x 5 0x 5x 7 Teachig Notes: The radicad ad the idex must be the same i order to add or subtract radicals. Sometimes it is ecessar to simplif radicals first to fid out if the ca be added or subtracte a a a a The quotiet rule ca be used i two was: or. b b b b Aswers:. a a. 5 x 8. a. 9 7 caot be subtracted 5. a. 5x x 0x x. a. x 5x x 5 Copright 0 Pearso Educatio, I ML-7
7 Mii Lecture 0.5 Multiplig With More Tha Oe Term ad Ratioalizig Deomiators Learig Objectives:. Multipl radical expressios with more tha oe term.. Use polomial special products to multipl radicals.. Ratioalize deomiators cotaiig oe term.. Ratioalize deomiators cotaiig two terms. 5. Ratioalize umerators.. Multipl. a. ( x ) x ( x ) ( 5)( 5) ( 7 5) e. ( )( ) f. ( x 5)( x 5). Ratioalize each deomiator. a. 5 x 5 x e. 5 f.. Ratioalize the umerator. x x Teachig Notes: To multipl radical expressios with more tha oe term use the distributive propert ad the FOIL metho To ratioalize the deomiator, multipl the umerator ad the deomiator b a radical of idex that produces a perfect th power i the deomiator s radica Radical expressios that ivolve the sum ad differece of the same two terms are called cojugates. To ratioalize a deomiator with two terms ad oe or more square roots, multipl the umerator ad deomiator b the cojugate of the deomiator. Aswers:. a. x 0. a. 5 x. x x8 5x 5 x x e. f. x 5 x e. 5 f. Copright 0 Pearso Educatio, I ML-7
8 Mii Lecture 0. Radical Equatios Learig Objectives:. Solve radical equatios.. Use models that are radical fuctios to solve problems. Solve.. a. x a x 5 0 e. x x 5 f. x 7. a. a 7 x 0 x 7 a a e. f. x 5 5 x Solve. Each of the followig examples will require squarig both sides twice.. a. x 8 x a 5 a 8 Solve.. a. ( 5x 7) x (5x ) ( 5x ) Teachig Notes: Whe solvig equatios with radicals, isolate the radical o oe side first. Raise both sides of the equatio to the power that is the idex of the radical i order to elimiate the radical. Sometimes this step must be doe a secod time to clear the equatio of all radicals. Alwas check all solutios for extraeous solutios. Aswers:. a. 0 o solutio 5 7 e. 9 f. 9. a. 8 e. 8 f. 0. a a. 5 0, 0 Copright 0 Pearso Educatio, I ML-7
9 Mii Lecture 0.7 Complex Numbers Learig Objectives:. Express square roots of egative umbers i terms of i.. Add ad subtract complex umbers.. Multipl complex umbers.. Divide complex umbers. 5. Simplif powers of i.. Write as a multiple of i. a Perform the idicated operatio. Write the result i the form a + bi. a. ( i) ( 7i) ( i) ( i) i ( i) ( i)(5 i) e.. Divide ad simplif the form a bi. i a. i i i. Simplif. a. i 0 i 0 i 7 i Teachig Notes: The imagiar uit i is defied as i where i. If b is a positive real umber, the b b( ) b i b. The set of all umbers i the form a bi, a is the real part, b is called the imagiar part of the complex umber a bi. Whe addig or subtractig complex umbers, add or subtract their real parts. The add or subtract their imagiar parts ad express the aswer as a complex umber. Studets eed to be remaied ofte to write their aswers i a + bi form. Aswers:. a. 9i i i 5. a. 7 i i i i i 5i 5 e.. a. or i or i. a. i i Copright 0 Pearso Educatio, I ML-7
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