Unit 6: Exponents and Radicals

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1 Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): - positive nd negtive whole numers. {,,,,,, 0,,,,,, } Rtionl Numers (Q): - numers tht cn e turned into frction, where, I, nd 0. - include ll Terminting or Repeting Decimls. - include ll Nturl Numers, Whole Numers nd Integers. - include n perfect roots (rdicls). ) Terminting Decimls: decimls tht stops ) Repeting Decimls: decimls tht repets in pttern nd goes on. 0.. c) Perfect Roots: - rdicls when evluted will result in either Terminting or repeting decimls, or 0. ± 0. frctions, where, I, nd ± 0... ± ± Irrtionl Numers (Q ): - numers tht CANNOT e turned into frction, where, I, nd 0. - include ll non-terminting, non-repeting decimls. - include n non-perfect roots (rdicls). ) Non-terminting, Non-repeting Decimls: - decimls tht do not repet ut go on nd on. π. 0. ) Non-Perfect Roots: rdicls when evluted will result in Non-Terminting, Non-Repeting decimls. ± ± ± Rel Numers (R): - n numers tht cn e put on numer line. - include ll nturl numers, whole numers, integers, rtionl nd irrtionl numers. Asolute Vlue : - the positive vlue of. () + Pge. Coprighted Griel Tng, B.Ed., BSc.

2 Pure Mth 0 Notes Eponents nd Rdicls In generl, we cn displ the reltionships etween ll tpes of rel numers in digrm. Rel Numers (R) Q Q I W N ) All Nturl Numers elong to the set of Whole Numer. N W ) All Nturl nd Whole Numers elong to the set of Integers. N nd W I c) All Nturl, Whole Numers nd Integers elong to the set of Rtionl Numers. N, W nd I Q d) Rtionl Numers nd Irrtionl Numers do NOT elong to ech other. (You cn hve oth tpes t the sme time). Q Q e) All Nturl, Whole Numers, Integers, Rtionl nd Irrtionl Numers elong to the set of Rel Numers. N, W, I, Q nd Q R Emple : Clssif the following numers... c.. d.. I, Q nd R N, W, I, Q nd R Q nd R Q nd R e. f.. g. h. Q nd R Q nd R Q nd R Q nd R i) 0. j) k) l) Q nd R N, W, I, Q nd R N, W, I, Q nd R Q nd R Coprighted Griel Tng, B.Ed., B.Sc. Pge.

3 Eponents nd Rdicls Pure Mth 0 Notes Inequlities Smols > Greter thn < Less thn lower upper lower < < upper lower nd upper Greter thn or equl to Less thn or equl to NOT Equl to Menings is etween the lower nd upper oundries (inclusive). is etween the lower nd upper oundries (eclusive). is less thn the lower oundr nd is greter thn the upper oundr (inclusive). < lower nd > upper is less thn the lower oundr nd is greter thn the upper oundr (eclusive). Emple : Grph the following inequlities. ) n > ) 0 0 c) <. d) m π. π 0 0 e) r f) < w 0 0 g) > nd 0 h) t 0 0 Pge. Coprighted Griel Tng, B.Ed., BSc.

4 Pure Mth 0 Notes Eponents nd Rdicls (AP) Emple : Grph the following inequlities. ) <. ) r. Sme s. Sme s r < nd r > 0 0 c) > nd 0 Sme s 0 < r 0 Emple : Convert the following decimls to frctions lgericll. ) 0. ). Let 0. (To cncel out the repeting Let. 0. decimls, we hve to move 00. the deciml plce to the right, which mens 0) (Move the deciml plces to the right will line up the repeting decimls) c) 0. d). Let 0. First, ignore the Let negtive sign. 0. (Move the deciml 000. plces to the right will line up the repeting 000. decimls) Put the negtive sign ck! 0 0 Put the negtive sign ck! First, ignore the negtive sign. (Move the deciml plce to the right will mke the repeting decimls pper right fter the deciml point.) Homework Assignments Regulr: pg. to # to,,,,, nd AP: pg. to # to, to, to, nd Coprighted Griel Tng, B.Ed., B.Sc. Pge.

5 Eponents nd Rdicls Pure Mth 0 Notes -: Evluting Irrtionl Numers Rdicls: - the result of numer fter root opertion. Rdicl Sign: - the mthemticl smol. Rdicnd: - the numer inside rdicl sign. inde n rdicl sign rdicnd Inde: - the smll numer to the left of the rdicl sign indicting how mn times numer (nswer to the rdicl) hs to multipl itself to equl to the rdicnd. squre root cue root fourth root fifth root To cll up the cue root or Choose Option for cue root higher root functions, press MATH Choose Option for higher root. But e sure to enter the numer for the inde first! Emple : Evlute the followings... c. d. ± ± ()() () ()() () ()()() ()()()() () ()()()() () ()()()()() A rdicl with n even inde lws hs two nswers (±), nd cn onl hve rdicnd greter thn or equl to 0 inside rdicl sign. A rdicl with n odd inde lws hs one nswer onl nd cn hve negtive rdicnd inside the rdicl sign. Emple : A formul v f v i + d cn e used to find the finl velocit (speed) of n ccelerted oject, where v f finl velocit, v i initil velocit, ccelertion, nd d distnce trvelled. An pple is thrown from the tll uilding 00 m high with n initil velocit of m/s. The ccelertion due to grvit is. m/s. Wht is the finl velocit of the pple s it reches the ground? Solve for v f : f Pge 00. v f? v i m/s d 00 m. m/s v v f f vi + d v ( ) + (.)( 00) v i + d v v f f + v f. m/s Coprighted Griel Tng, B.Ed., BSc.

6 Pure Mth 0 Notes Eponents nd Rdicls Estimting Squre Roots. Estimting Squre Roots GREATER thn :. Group the rdicnd two digits strting directl to the LEFT of the deciml plce. The digit 0 m e dded to the eginning of the rdicnd if there re n odd numer of digits.. Estimte ech group of two digits finding the squre root of the nerest lower squre numer.. Estimting Squre Roots LESS thn :. Group the rdicnd two digits strting directl to the RIGHT of the deciml plce. The digit 0 m e dded to the end of the rdicnd if there re n odd numer of digits.. Estimte ech group of two digits finding the squre root of the nerest lower squre numer. Emple : Estimte. Then, find the pproimted vlue to the fifth deciml plce using clcultor with onl positive roots.... c. 0. d Actul.0 Actul.00 Actul 0. Actul 0.0 Emple : Evlute estimting, then, find the pproimted vlue to the fifth deciml plce using clcultor with onl positive roots.. ( )( 0). 0 ( ) c. 0 0 () 0 (). Actul.0 Actul.0 Actul. Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

7 Eponents nd Rdicls Pure Mth 0 Notes Emple : Evlute the followings using onl positive roots... c. d Emple : Evlute the followings using onl positive roots. Verif using clcultor.. +. () + () () 0 + Emple : Evlute the followings using onl positive roots. Verif using clcultor ( ) ( ) ( 0.0 ) ( 0.) 0. - Homework Assignments Regulr: pg. to # to (odd), to (no estimtes), to, to,, nd AP: pg. to # to (even), to (no estimtes), to, to, nd Pge 0. Coprighted Griel Tng, B.Ed., BSc.

8 Pure Mth 0 Notes Eponents nd Rdicls -: Simplifing Rdicls where 0 nd 0 where 0 nd > 0 Entire Rdicls: - rdicls tht hve no coefficient in front of them. Emples: nd Mied Rdicls: - rdicls tht hve coefficients in front of them. - the coefficient is the squre root of the perfect squre fctor of the rdicnd. Emples: nd To convert n entire rdicl to mied rdicl, find the lrgest perfect squre fctor of the rdicnd nd write its root s coefficient of the remining rdicnd fctor. Emple : Simplif the followings. (Convert them to mied rdicls) c. d. To convert mied rdicl to n entire rdicl, squre the coefficient nd multipl it to the rdicnd. Emple : Write the followings s entire rdicls... c Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

9 Eponents nd Rdicls Pure Mth 0 Notes Emple : Order,, nd from lest to gretest. < < 0 0 < < Emple : Simplif... ( )( ) Rtionliztion: - turning rdicl denomintor into nturl numer denomintor multipling frction of the rdicl over itself. Emple : Simplif Pge 0. Coprighted Griel Tng, B.Ed., BSc. c. OR 0 0 0

10 Pure Mth 0 Notes Emple : Simplif... 0 c. Need to find perfect cue fctor of the rdicnd. We cn hve negtive perfect cue fctor. Emple : Write the followings s entire rdicls... c. 0 We need to cue the ( ) coefficient nd multipl it into the rdicnd Eponents nd Rdicls d. Need to find perfect fourth fctor of the rdicnd. d. We need to tke the coefficient to the fourth power nd multipl it into the rdicnd. 0 (AP) Emple : Simplif.. We hve to multipl the cue root of the squre of the rdicnd to form perfect cue.. (AP) Emple : Solve for... ( ) Coprighted Griel Tng, B.Ed., B.Sc. Pge 0. 0 c. - Homework Assignments Regulr: pg. to 0 # to (odd), to, to (even), to, AP: pg. to 0 # to (even), to, to (odd), to

11 Eponents nd Rdicls Pure Mth 0 Notes -: Opertions with Rdicls Adding nd Sutrcting Rdicls: Convert n entire rdicls into mied rdicls first. Then, comine like terms (rdicls with the sme rdicnd) dding or sutrcting their coefficients. Emple : Simplif the followings ( ) ( ) ( ) Multipling Rdicls: When multipling two mied rdicls, multipl the coefficients first, nd then multipl the rdicnds. Simplif ech term fterwrds if necessr. Emple : Simplif the followings.. ( + ). ( ) c. ( + )( ) ( + ) ( ) ( + )( ) ( ) ( ) ( ) ( ) 0 () 0 d. ( )( + ) e. ( ) ( )( + ) + 0 ( ) + 0() + ( ) ( )( ) () + ( ) + Pge 0. Coprighted Griel Tng, B.Ed., BSc. + +

12 Pure Mth 0 Notes Eponents nd Rdicls Conjugtes: - inomils tht hve the ect sme terms opposite signs in etween. + c d nd c Emple: ( + ) nd ( ) ( ) ( d ) Multipling Conjugte Rdicls: The result of multipling conjugte rdicls is ALWAYS Rtionl Numer (the rdicl terms will lws cncel out). Emple : Simplif the followings.. ( + )( ). ( )( + ) ( + )( ) ( )( + ) ( ) ( ) ( ) 0 Notice the middle two terms lws cncel out! Rtionlizing Binomil Rdicl Denomintors: Multipl the rdicl epressions frction consist of the conjugte of the denomintor over itself Emple : Simplif the followings.. + ( + ) ( ) ( ) ( ) ( + ) ( ) ( + ) ( ) ( ) ( ) ( ) ( ) Emple : A rectngle hs perimeter of 0 + nd its width is 0. Wht is the length of this rectngle? ( 0 + ) l ( 0 ) P ( l + w) Sutrcting l ( 0 + ) ( 0 ) w 0 P 0 + P rcket! l + w ( 0 ) ( 0 + ) l Switch signs in the second rcket! P w l length 0 + Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

13 Eponents nd Rdicls Pure Mth 0 Notes Emple : Find the volume of clinder tht hs rdius of, nd its height is r h ( ) V V V V V V V V πr π π π π π π π h ( ) ( ) ( )( )( ) ( ( ( ) () ) ( )( ) ( ( 0( ) ( )( 0 ) ( 0 0) Volume π 0π 0 Emple : A prllelogrm hs n re of +. Clculte the mesure of its height if the se is h A ( ). ( + ) A h h h h A ( + ) ( ) ( + ) ( + ) ( ) ( + ) Height () + Emple : Simplif (AP) c. ( + )( ) ( ) + 0( ) 0( ) ( ) ( ) + ( ) - Homework Assignments Regulr: pg. to # to (odd), to 0, to, 0 AP: pg. to # to 0 (even), to 0, to 0 Pge 0. Coprighted Griel Tng, B.Ed., BSc. + +

14 Pure Mth 0 Notes Eponents nd Rdicls -: Reviewing the Eponent Lws power m eponent se Eponentil Lws m ( ) ( m )( n ) m + n n ( ) n n mn ( m ) n mn 0 () m m m m m m Emple : Simplif. Epress ll nswers in positive eponents onl.. (c d )(c d ). c + c d d c. ( ) ()( )( ) d. ( ) ( ) ( ) 0 ( )( ) ( m n ) ( ) ( m n ) e. (m n ) (m n ) f. m m m () n n n n 0 0 When reciprocting n entire rcket, do NOT mess with its content. n p q p q p p p q p q q q p q p q ( ) ( ) p q Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

15 Eponents nd Rdicls Pure Mth 0 Notes g. ( ) ( ) + ( ) h. ( h k ) ( h k ) ( h k ) ( h k ) ( h k ) ( h k ) h ( h k )( h k ) h 0 0 ( ) ( ) ( ) k k h k Emple : In stronom, one light er is the distnce light cn trvel in one er. Light hs constnt speed of 0 km/s in the vcuum of spce.. Clculte the distnce of one light er.. The closest str to the Sun, Alph Centuri, is. 0 km. How mn light ers is it to our sun?. One Light Yer ( 0 km/s)( ds/r)( hr/d)(0 min/hr)(0 s/min) EE, Emple : Solve for.. ( )( ) ± 0 (AP) Emple : Simplif.. ( m ) () m +. m ( ) ( ) m+ ( m+ ) ( ) m+ ( ) m+.0 0 km/r + ( ) + (AP) c. Pge 0. Coprighted Griel Tng, B.Ed., BSc... 0 km.0 0 km / r ( ) + + ( + ) ( + ) ( ) ( + ) ± light ers - Homework Assignments Regulr: pg. # to (even), 0 to,,,, AP: pg. # to (odd), 0 to

16 Pure Mth 0 Notes Eponents nd Rdicls -: Rtionl Eponents m n n m The inde of the rdicl is the denomintor of the frctionl eponent. Emple : Evlute... ( ) e. ( ) f. Emple : Evlute using clcultor.. ( ). ( ). c. ( ) 0 d. ( ) 0 ( ) g. c.. ( ) Emple : Write the followings using eponents... c. ( ) ( ) d. ( ) ( ) ( ) Coprighted Griel Tng, B.Ed., B.Sc. Pge.

17 Eponents nd Rdicls Pure Mth 0 Notes Pge. Coprighted Griel Tng, B.Ed., BSc. Emple : Evlute, if possile.. ( )( ). c. ( ) Emple : Write the following epressions using eponents... c. ( )( ) d. ( ) (AP) Emple : Solve for... + c. - Homework Assignments Regulr: pg. to # to (odd), to, to (no estimtes), 0, AP: pg. to # to (even), to, to (no estimtes), 0 to + ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) + + +

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