Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd Ex : Evlute ech. If ot rel, write ot rel. ) b) 6 c) 6 + 6 d) 6 + 6 e) 9 My ties studets believe tht positive. Exie the followig. 6 ( ) 6, ot 6 6, which is ot rel. However, the pricipl squre root is lwys I geerl: Therefore, we will lwys ssue tht vribles represet positive ubers i order to void usig bsolute vlue sigs. II Other Types of Roots b es tht b If is eve, the d b ust be positive. If is odd, d b c be y rel ubers. idex If o idex is writte, the root is ssued to be squre root.
Ex : Evlute ech. If ot rel, write ot rel. b c ) ) 6 ) 6 d) 7 e) 0.0 f ) III The Product d Quotiet Rules of Rdicls If ll expressios represet rel ubers, b b d b b d ( b 0) b b b b Note: These properties re for ultiplictio d divisio. Siilr stteets re ot true for dditio or subtrctio. ( + b + b, for exple) Ex : Use the product or quotiet rules of rdicls (if you c) to write s oe rdicl. Siplify, if possible. ) 0 b) c ) IV Siplifyig Squre Root Rdicls A squre root is siplified whe its rdicd hs o fctors other th tht re perfect squres. Reeber:, if is ssued to be positive. We will ssue ll vribles represet positive vlues.
Ex : Use fctorig d the product (d/or quotiet)rule to siplify ech. ) x b) b 7 c) x x d) V Additio d Subtrctio of Squre Roots Two or ore squre roots c be cobied if they hve the se rdicd. Such rdicls re clled like rdicls. Soetie oe or ore rdicl ust be siplified i order to cobie. Ex : Siplify d cobie where possible. ) + 6
b) c ) 6 VI Rtiolizig Deoitors The process of rewritig squre root rdicl expressio s equivlet expressio i which the deoitor o loger cotis y rdicls is clled rtiolizig the deoitor. First, siplify y rdicls. Secodly, ultiply the uertor d deoitor by the rdicl fctor tht reis. Ex 6: Siplify by rtiolizig the deoitor. )
b) c) VII Cojugtes Rdicl expressios tht ivolve the su d differece of the se two ters re clled cojugtes. Exples re + d or x d + x. The product of two cojugtes will coti o rdicls! ( + b )( b ) ( ) ( b ) b I rdicl expressios with bioil (two ters) i the deoitor, to rtiolize the deoitor, ultiply uertor d deoitor by the cojugte of the deoitor. Ex 7: Rtiolize d siplify ech. ) b) +
VIII Rtiol Expoets Exie: ( ) ust be equivlet. d Sice both d squred equl, they Defiitio of If represets rel uber, where is iteger, the. Ex : Evlute ech, if it exists. ) 9 b) c) 6 The deoitor of the rtiol expoet becoes the idex of the rdicl. The textbook d olie hoework y use regulr frctio br for rtiol expoet or slsh frctio br. / d) e) ( ) Exie: ( ) d ( ) Therefore: ( ) or 6
Defiitio of If represets rel uber d is ( ) or positive rtiol uber,, the. It c be evluted or siplified by fidig the power first, the the root or by fidig the root first, the the power. Becuse you will ot hve clcultor o quizzes or your first ex, I recoed fidig the root first, the rise to the expoet power. The uertor is the expoet. or ( ) The deoitor is the idex. Ex 9: Evlute, if possible. ) b) c) 6 6 Ex 0: Evlute, if possible. ) b) ( ) 7
Ex : Use the properties of expoets to siplify. c) d) 6 b 9 ( x ) 6 6 6 e) (6 x y ) ( x y ) Ex : A rectgle below hs the give width d legth. Fid the perieter (usig rdicls s eeded) d the re (usig rdicls s eeded) of this rectgle. Siplify ech. 67 Soe theticl odels y be equtios tht hve rdicl expressios. Ex : Suppose E x +. odels the uber of elderly Aerics ges 6-, i illios, for x uber of yers fter 00. Project the uber of Aerics ges 6-, i illios, i 00 d 00. Express the icrese i uber of elderly Aerics fro 00 to 00.