Roots, Radicals, and Complex Numbers


 Dwight Thornton
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1 Chpter 8 Roots, Rils, Comple Numbers Agel, Itermeite Algebr, 7e Lerig Objetives Workig with squre roots Higherorer roots; ris tht oti vribles Simplifig ril epressios Agel, Itermeite Algebr, 7e Squre Roots (terms efiitios) Ril epressio: A mthemtil lultio or formul ombiig umbers /or vribles usig sums, ifferees, prouts, quotiets (iluig frtios), epoets, roots, logrithms, trig futios, pretheses, brkets, futios, or other mthemtil opertios. Agel, Itermeite Algebr, 7e
2 Squre Roots (terms efiitios) Ri: The vlue isie the ril sig. The vlue ou wt to tke the root of. Agel, Itermeite Algebr, 7e Squre Roots (terms efiitios) Fiig the squre root of give umber is the reverse proess of squrig umber. Whe fiig the squre root of umber, we re etermiig wht umbers, whe multiplie b themselves, result i the give umber. Emples: If 7, the If 7, the ( 7)( 7) 9 Agel, Itermeite Algebr, 7e 5 Squre Roots (terms efiitios) is re the "squre root of." The is lle the ril sig. The epressio isie the ril sig is lle the ri. The etire epressio, iluig the ril sig ri, is lle the ril epressio. Agel, Itermeite Algebr, 7e 6
3 Squre Roots (terms efiitios) The positive or priipl squre root of positive umber is writte s. The egtive squre root is writte s . b if b Also, the squre root of 0 is 0, writte 0 0. Note tht the priipl squre root of positive umber,, is the positive umber whose squre equls.. Wheever the term squre root is use i this book, the positive or priipl squre root is met to be use. Agel, Itermeite Algebr, 7e 7 Squre Roots (terms efiitios) The ie tells the root of the epressio. Sie squre roots hve ie of, the ie is geerll ot writte i squre root. mes Emple: 5 5 (sie ) 9 9 (sie ) 6 6 Agel, Itermeite Algebr, 7e 8 Squre Roots (terms efiitios) Squre roots of egtive umbers re ot rel umbers. Squre roots of egtive umbers re lle imgir umbers. 5? There is o umber multiplie b itself tht will give ou 5. Imgir umbers will be isusse lter i this hpter Agel, Itermeite Algebr, 7e 9
4 Squre Roots (terms efiitios) A perfet squre is the squre of turl umber.,, 9, 6, 5, 6 re the first si perfet squres. A rtiol umber is oe tht be writte i the form, where b re itegers, b b 0. Rel umbers tht re ot rtiol umbers re lle irrtiol umbers. As eimls, irrtiol umbers re orepetig, otermitig eimls. Agel, Itermeite Algebr, 7e 0 Perfet Squres Agel, Itermeite Algebr, 7e Approimtig Squre Roots Here re the steps to pproimte squre root:. Pik perfet squre tht is lose to the give umber. Tke its squre root.. Divie the origil umber b this result.. Tke the rithmeti me of the result of I the result of II b ig the two umbers iviig b (this is lso lle ʺtkig vergeʺ).. Divie the origil umber b the result of III. 5. Tke the rithmeti me of the result of III the result of IV. 6. Repet steps IVVI usig this ew result, util the pproimtio is suffiietl lose. Agel, Itermeite Algebr, 7e
5 Approimtig Squre Roots Emple : Approimte the.. 5 is lose to 5 5. /5.. (5 +.)/.7. / ( )/ / So.69 Agel, Itermeite Algebr, 7e Rtiol, Irrtiol, Imgir Numbers Rtiol umbers: the solutio is iteger. 9 Irrtiol umbers: the solutio is ot iteger. Imgir umbers: 9 Agel, Itermeite Algebr, 7e Agel, Itermeite Algebr, 7e 5
6 Cube Fourth Roots is re the ube root of. is re the fourth root of. b if b b if b 8 sie 8 8 sie ( )( )( ) 8 8 sie 8 Agel, Itermeite Algebr, 7e 6 Eve O Iies Eve Iies The th root of,, where is eve ie is oegtive rel umber, is the oegtive rel umber b suh tht b. 8 sie sie0 00 Agel, Itermeite Algebr, 7e 7 Eve O Iies O Iies The th root of,, where is o ie is rel umber, is the rel umber b suh tht b. 6 sie sie () Agel, Itermeite Algebr, 7e 8
7 Cube Fourth Roots Note tht the ube root of positive umber is positive umber the ube root of egtive umber is egtive umber. The ri of fourth root (or eve root) must be oegtive umber for the epressio to be rel umber. Agel, Itermeite Algebr, 7e 9 8. Simplifig Rils Agel, Itermeite Algebr, 7e 0 Defiitios A perfet squre is the squre of turl umber.,, 9, 6, 5, 6 re the first si perfet squres. Vribles with epoets m lso be perfet squres. Emples ilue,( ) ( ). A perfet ube is the ube of turl umber., 8, 7, 6, 5, 6 re the first si perfet ubes. Vribles with epoets m lso be perfet ubes. Emples ilue, ( ) ( ). Agel, Itermeite Algebr, 7e
8 Perfet Powers This ie be epe to perfet powers of vrible for ri. The ri is perfet power whe is multiple of the ie of the ri. A quik w to etermie if ri is perfet power for ie is to etermie if the epoet is ivisible b the ie of the ril. Emple: 5 0 Sie the epoet, 0, is ivisible b the ie, 5, 0 is perfet fifth power. Agel, Itermeite Algebr, 7e Prout Rule for Rils For oegtive rel umbers b b b, Emples: Agel, Itermeite Algebr, 7e Prout Rule for Rils To Simplif Rils Usig the Prout Rule. If the ri otis oeffiiet other th, write it s prout of the two umbers, oe of whih is the lrgest perfet power for the ie.. Write eh vrible ftor s prout of two ftors, oe of whih hihis the lrgest perfet power of fthe vrible ibl for the ie.. Use the prout rule to write the ril epressio s prout of rils. Ple ll the perfet powers uer the sme ril.. Simplif the ril otiig the perfet powers. Agel, Itermeite Algebr, 7e
9 Prout Rule for Rils Emples: b b b b b b b *Whe the ril is simplifie, the ri oes ot hve vrible with epoet greter th or equl to the ie. Agel, Itermeite Algebr, 7e 5 Quotiet Rule for Rils For oegtive rel umbers, b 0 b b b, Emples: Simplif ri, if possible Agel, Itermeite Algebr, 7e 6 Quotiet Rule for Rils More Emples: b 6 b 8 8 6b 8 6b 8 6b b 8 8 Agel, Itermeite Algebr, 7e 7
10 8. 5 Aig, Subtrtig, Multiplig Rils Agel, Itermeite Algebr, 7e 8 Like Rils Like rils re rils hvig the sme ris. The re e the sme w like terms re e. Emple: z + 0 z 5 z 8 z Cot be simplifie further. Agel, Itermeite Algebr, 7e 9 Aig & Subtrtig To A or Subtrt Rils. Simplif eh ril epressio.. Combie like rils (if there re ). Emples: (6 ) Agel, Itermeite Algebr, 7e 0
11 CAUTION! The prout rule oes ot ppl to itio or subtrtio! b b + b + b Agel, Itermeite Algebr, 7e Multiplig Rils Multipl: ( 5 ) 5 (8 + 5)(6 ) Use the FOIL metho. ( + 6)( 6) ( ) Notie tht the ier outer terms el. Agel, Itermeite Algebr, 7e Multiplig Rils More Emples: 8 ( 7 ) z z z z z z z 7 5 Agel, Itermeite Algebr, 7e
12 Diviig Rils Agel, Itermeite Algebr, 7e g Rtiolizig Deomitors Emples: To Rtiolize Deomitor 6 Multipl both the umertor the eomitor of the frtio b ril tht will result i the ri i the eomitor beomig perfet power. C t b i lifi f th Agel, Itermeite Algebr, 7e 5 Emples: r pr q r r pq r r pq r r r pq r pq Cot be simplifie further ) ( Simplifig Rils Simplif b rtiolizig the eomitor: 5 + Agel, Itermeite Algebr, 7e ) )( ( ) )( ( + +
13 Simplifig Rils A Ril Epressio is Simplifie Whe the Followig Are All True. No perfet powers re ftors of the ri ll epoets i the ri re less th the ie.. No ri otis frtio.. No eomitor otis ril. Agel, Itermeite Algebr, 7e 7 Simplifig Rils Simplif: ( r + ) 6 5 ( r + ) 5 5 ( r + ) ( r + ) 6 5 ( 5 6) ( 5 ) ( 5 6) ( 0 6) 5 6 ( r + ) ( r + ) ( r + ) 5 6 ( r + ) Agel, Itermeite Algebr, 7e Solvig Ril Equtios Agel, Itermeite Algebr, 7e 9
14 Ril Equtios A ril equtio is equtio tht otis vrible i ri To solve ril equtios suh s these, both sies of the equtio re squre ( ) ( + ) ( ) ( 7) 0 ; 7 ( ) Agel, Itermeite Algebr, 7e 0 Etreous Roots I the previous emple, etreous root ws obtie whe both sies were squre. A etreous root is ot solutio to the origil equtio. Alws hek ll of our solutios ito the origil equtio. 0 ; 7 Chek: Chek: FALSE! Agel, Itermeite Algebr, 7e Two Squre Root Terms To solve equtios with two squre root terms, rewrite the equtio, if eessr so tht there is ol oe term otiig squre root o eh sie of the equtio. Solve the equtio: 5 + ( 5) ( + ) Chek: 7 (7) Agel, Itermeite Algebr, 7e
15 Noril Terms Solve the equtio: b b + ( b ) ( b + ) b 6 8 b + + b + b 0 8 b + ( b 0 ) ( 8 b + ) b + 0b (b + ) b + 0b b + 6 b 88b ( b 8 )( b ) 0 b 8, b Chek: b 8 b b Not solutio. b + 9 Agel, Itermeite Algebr, 7e Summr To Solve Ril Equtios. Rewrite the equtio so tht oe ril otiig vrible is isolte o oe sie of the equtio.. Rise eh sie of the equtio to power equl to the ie of the ril.. Combie like terms.. If the equtio still otis term with vrible i ri, repet steps through. 5. Solve the resultig equtio for the vrible. 6. Chek ll solutios i the origil equtio for etreous solutios. Agel, Itermeite Algebr, 7e 8.7 Rtiol Epoets Agel, Itermeite Algebr, 7e 5
16 Chgig Ril Epressio A ril epressio be writte usig epoets b usig the followig proeure: Whe is oegtive, be ie. Whe is egtive, must be o ( ) ( 7 ) 9 + 7z + z Agel, Itermeite Algebr, 7e 6 Chgig Ril Epressio Whe is oegtive, be ie. Whe is egtive, must be o. Epoetil epressios be overte to ril epressios b reversig the proeure. 5 b b 5 Agel, Itermeite Algebr, 7e 7 Simplifig Ril Epressios This rule be epe so tht rils of the m form be writte s epoetil epressios. For oegtive umber, itegers m, Power m m ( ) m Ie b b ( 8 9) ( ) Agel, Itermeite Algebr, 7e 8
17 Rules of Epoets The rules of epoets from Setio 5. lso ppl whe the epoets re rtiol umbers. For ll rel umbers b ll rtiol umbers m, Prout rule: Quotiet rule: Negtive epoet rule: m m + m m m, 0, 0 m Agel, Itermeite Algebr, 7e 9 Rules of Epoets For ll rel umbers b ll rtiol umbers m, Zero epoet rule: Risig power to power: Risig prout to power : Risig quotiet to power : 0, 0 m m ( ) m m m ( b ) b b m b m m, b 0 Agel, Itermeite Algebr, 7e 50 Rules of Epoets Emples:.) Simplif / /5. 5 ( ) ( 5) ( 5 0) ( 0) ) Simplif ( /5 ) /. ( 5)( ) ( ) ) Multipl /9 ( /9 ). ( 9) + ( 9) ( 9) Agel, Itermeite Algebr, 7e 5
18 Ftorig Epressios Emples:.) Ftor / 5/. The smllest of the two epoets is /. / 5/ / ( 5/(/) ) / ( / ) / ( ) Origil epoet Epoet ftore out.) Ftor / + /. The smllest of the two epoets is /. + / + / / ( /(/) ) / ( ) Origil Epoet epoet ftore out Agel, Itermeite Algebr, 7e 5
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