= 2, 3, 4, etc. = { FLC Ch 7. Math 120 Intermediate Algebra Sec 7.1: Radical Expressions and Functions

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1 Math 120 Itermediate Algebra Sec 7.1: Radical Expressios ad Fuctios idex radicad = 2,,, etc. Ex 1 For each umber, fid all of its square roots Ex 2 1 Simplify eve must be 0 odd ay real umber Ex Simplify. (Assume all variables represet ay real umber.) AAVRARN 16t 2 (a + ) 2 y 8 y 8 (7b) 9 7b 9 6 ( ) 6 7 ( ) 7 9 6y + y 2 ( ) = {, if is eve ( ), if is odd Ex Simplify. Assume that o radicads were formed by raisig egative quatities to eve powers. (Assume that variables represet ay positive real umber.) AAVR+N 2t 2 (7xy) 2 (7xy) 6y 6 a 1 (x + ) 10 Page 1 of 1

2 Ex Fid the domai of each fuctio. f(x) = + x g(x) = + x 6 h(x) = + x s(x) = 1x + x Evaluate: f(0), f(1), ad f( 2) Sec 7.2: Ratioal Numbers ad Expoets Recall: Expoetial Rules x a x b = x a+b x a x b = xa b, x 0 (x a ) b = x ab (xy) a = x a y a ( x y )a = xa y a x a = 1 x a, x 0 x 0 = 1, x 0 What does x 1/2 mea? x 1/? Examples: 9 1/2 = ( 2 ) 1/2 = 1 = 8 1/ 16 1/2 27 1/ 9 1/2 12 1/ Coclusio: ( ) 1/2 = ad ( ) 1/ = I geeral, a 1/ = a. Note: If is eve, a must be 0. If is odd, a ca be aythig. a m/ = a m = ( a) m if a exists. Page 2 of 1

3 Ex 6 Write a equivalet expressio usig radical/expoetial otatio, ad if possible, simplify. Which umbers are ratioal? Irratioal? t 1/ (a 2 b) 1/ 7/2 (9y 6 ) / cd Ex 7 Rewrite but do ot simplify. ( 2xy 2 z) x 2 y z 7 Ex 8 Simplify. Do ot use egative expoets i aswer. Which umbers are ratioal? Irratioal? 1/ 1/8 8 7/11 8 2/11 (/ ) /7 (27 /9 x 1/ y 2/ ) /2 Ex 9 12 a Simplify. Preset aswers i radical form. ( ab) 1 8 (x) 2 ab 1 6 x ( x 2 y ) 12 2a Several problems from Writig Expressios as Powers of x hadout page 8. Page of 1

4 Sec 7.: Multiplyig Radical Expressios Rules a b = ab ad ab = a b as log as a ad b are real umbers. Ex 10 8 Simplify y 2a Ex 11 Multiply. 6 2 x a x + a 1 98 Ex 12 Fid a simplified form of f(x) = 2x 2 + 8x + 8 ad g(x) = x 2 + 8x + 8 Ex 1 Simplify. Assume that o radicads were formed by raisig egative umbers to eve powers. a) x 8 y 7 b) 2a 7 b 11 c) 810x 9 d) 2 Page of 1

5 e) ( 10x 2 y )( 20x 2 y 6 ) f) a (b c) a 7 (b c) Ex 1 Simplify usig the laws of expoets. 2 1/10 2/ 2/ / 9 2/ (9k 2 m ) 1 2 ( xy 1/ z 1/2 ) 7 ( 8x 2 y) Express aswer i expoetial form Practice Problems-box ALL aswers 1) Simplify. Assume that each variable ca represet ay real umber. a) 6t 2 b) c 2 + 1c + 9 c) (c + 7) 2) Write a equivalet expressio usig radical/expoetial otatio. a) ( ab) b) (16a 6 ) / ) Simplify. Do ot use egative expoets i aswer. a) (x 2/ ) / b) 7 1/ 7 1/2 ) Simplify. Write all aswers i radical otatio. Assume that all variables represet oegative umbers. a) 20x y 2 b) x b 9xb 2 Page of 1

6 Ex 1 Rule: For ay real umbers a Sec 7.: Dividig Radical Expressios Simplify. Assume all variables represet positive umbers. AAVR+N a) b) 2a 2a9 b 6 b 1 c ad b, b 0, a b = a. b Ex 16 Divide ad if possible, simplify. AAVR+N a) b) 7ab 6a 11 b 28 2ab 2 Ex 17 Simplify. Assume all variables are oegative. ( 2)( 2) ( ) ( x) 2 x 2x Ex 18 Ratioalize each deomiator. AAVR+N a) b) Page 6 of 1

7 c) d) 21x 2 y 7 7xy 6a 2 b Ex 19 Simplify. For all but the first two problems, write fial aswers i expoetial form. AAVR+RN 7ab 6 x yz 7 1 9p 8/9 8 7/9 ax 7/8 z 8 6 2/ 6 / (x 2/ y / ) 1/2 Note: O quizzes ad exams must kow whe to use expoets vs radicals. Ex 20 Sec 7.: Expressios Cotaiig Several Radical Terms Add/subtract. Assume all variables represet oegative real umbers. AAVRNRN a) b) c) x 2x Page 7 of 1

8 Ex 21 Multiply. AAVRNRN a) 17( 2 17) b) x( x 2 81x 2 ) c) ( 2)(2 + 2) d) Let f(x) = x 2. Fid f( x ). Ex 22 Ratioalize each deomiator. a) PP b) c) As: x 2x z d) e) f) Test? x 6 x 6 2x + r + s r + s Ex 2 Simplify. AAVRNRN a) b) c) Provide forms of the aswer. b b xy 2 z x yz 2 x 2 x 6 expoetial, radical ot ratioalized, rad ratl Page 8 of 1

9 Sec 7.6: Solvig Radical Equatios Def A radical equatio is a equatio i which the variable appears i a radicad. Examples: 2x + 1 = a 2 = 7 x + 1 = 6 x The Priciple of Powers If a = b, the a = b for ay expoet. Warig: The coverse is ot true. **Always start by isolatig the radical ad check for extraeous solutios.** WILL NOT BE REMINDED TO ON EXAMS Ex 2 Solve. a) 2x 1 = 2 b) x 2 + = 2 c) x 2 + = 7 d) x 1/ = 9 e) y = f) 2x + = 2 g) x = x 1 + h) 2t 7 = t 12 i) 6x + 7 x + = 1 Page 9 of 1

10 Ex 2 If g(x) = x x, fid ay x for which g(x) =. Show a check. Sec 7.8: The Complex Numbers Ex Solve x 2 1 = 0 ad x = 0. Def of the Number i i is the uique umber for which i = 1 ad i 2 = 1. Note: i 1!!! Defs A imagiary umber is a umber that ca be writte i the form a + bi, where a ad b are real umbers ad b 0. A complex umber is ay umber that ca be writte i the form a + bi where a ad b are real umbers. Note: a ad b ca both be 0. The real part of a complex umber is a. The imagiary part is b. Cojugate of a Complex Number The cojugate of a complex umber a + bi is a bi, ad the cojugate of a bi is a + bi. Ex 26 Express i terms of i Ex 27 Circle all irratioal umbers. Box all oreal, complex umbers. Double uderlie all ratioal umbers e ( 2) + 9 2π 16 ( 2) π Page 10 of 1

11 Cycle of i Ex 28 Perform the idicated operatio ad simplify. Write each aswer i a + bi form. Idetify the real ad imagiary parts. a) (8 + 7i) (2 + i) b) 7i( 8i) c) 6 7 d) ( + i)( i) e) (1 + 2i)(1 2i) f) ( + 2i) 2 g) h) i) 26 7i + i 6i + i Page 11 of 1

12 j) k) l) m) Assume x i 9 i i 78 i 8 + i 0 18x x Sec 7.7: The Distace ad Midpoit Formulas ad Other Applicatios The Pythagorea Theorem I ay right triagle, if a ad b are the legths of the legs ad c is the legth of the hypoteuse, the a 2 + b 2 = c 2. Hypoteuse c a Leg 90 b Leg The Priciple of Square Roots For ay oegative real umber, if x 2 = the x = or x =. Ex 29 (# 20) How log is a guy wire if it reaches from the top of a 1-ft pole to a poit o the groud 10 ft from the pole? Legths Withi Isosceles ad Right Triagles The legth of the hypoteuse i a The legth of the loger leg i a isosceles right triagle is the legth right triagle is the legth of the shorter leg times. of a leg times 2. The hypoteuse is twice as log as the shorter leg. a a 2 a 0 2a a a 60 Page 12 of 1

13 Ex 0 For each triagle, fid the missig legth(s). Give a exact aswer ad, where appropriate, a approximatio to decimal places. a) (# 2) 18.8 b) (# ) 1. 1? 1? 1? 19? 1? The Distace Formula The distace d betwee ay two poits (x 1, y 1 ) ad (x 2, y 2 ) is give by d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. The Midpoit Formula If the edpoits of a segmet are (x 1, y 1 ) ad (x 2, y 2 ), the the coordiates of the midpoit are ( x 1+x 2 2, y 1+y 2 ). Note: To locate the midpoit, average the x-coordiates ad average the y-coordiates. 2 (x 2, y 2 ) (x 1, y 1 ) ( x 1 + x 2 2, y 1 + y 2 ) 2 Ex 1 (# ) Fid the distace betwee the pair of poits ( 1, ) ad (, ). Do two ways. PT the DF. Ex 2 (# 7) Fid the midpoit of the segmet with edpoits (, 2 ) ad (1 8, ). Page 1 of 1

14 Practice Problems (sec 7.2) (12x 1/2 ) 2/ (6xy 12/ ) 1/6 (8x 2 y ) 1/ (17x 2/ y 1/ z 1/ ) (1x y / z 1/6 ) 16i i 20 (Sec 6.) Terrel bicycles 10 mph with o wid. Agaist the wid, he bikes 12 miles i the same amout of time that it takes him to bike 8 miles with the wid. Set up a equatio or a system of equatios to fid the speed of the wid. Circle your equatio(s). Next, solve ad circle your aswer. Page 1 of 1

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3. Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is