Surds, and other roots - Math Centre

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1. IntroductionInthisunitwearegoingtoexplorenumberswrittenaspowers,andperformsomecalculationsinvolvingthem. Inparticular, wearegoingtolookatsquarerootsofwholenumberswhichproduceirrationalnumbers—thatis,numberswhichcannotbewrittenasfractions.Thesearecalledsurds.2. Powers and rootsWeknowthat2cubedis 2 × 2 × 2,andwesaythatwehave2raisedtothepower3,ortotheindex3. Aneasywayofwritingthisrepeatedmultiplicationisbyusinga‘superscript’,sothatwewouldwrite 2 3 :2 3 = 2 × 2 × 2 = 8 .Similarly,4cubedis 4 × 4 × 4,andequals64.Sowewrite4 3 = 4 × 4 × 4 = 64 .Butwhatifwehavenegativepowers?Whatwouldbethevalueof 4 −3 ?Tofindout,weshalllookatwhatweknowalready:4 3 = 4 × 4 × 4 = 64 ,4 2 = 4 × 4 = 16 ,4 1 = 4 = 4 ,andso 4 0 = 4 ÷ 4 = 1(becausetogettheansweryoudividethepreviousoneby4).Nowlet’scontinuethepattern:4 −1 = 1 ÷ 4 = 1 4 ,4 −2 1= 4 = 1 , 164 −3 1= 16 = 1 64and 164 = 1/43 . Soanegativepowergivesthereciprocalofthenumber—thatis,1overthenumber.Thus 4 −2 = 1/4 2 = 1 16 ,and 4−1 = 1/4 1 = 1 4 .Similarly,3 −2 = 1 3 2 = 1 9and5 −3 = 1 5 3 = 1125 .Acommonmisconceptionisthatsincethepowerisnegative,theresultmustbenegative: asyoucansee,thisisnotso.Nowweknowthat 4 0 = 1and 4 1 = 4,butwhatis 4 1/2 ?c○mathcentreJune23,2009 www.mathcentre.ac.uk 2 mc-TY-surds-2009-1
Usingtherulesofindices,weknowthat 4 1/2 × 4 1/2 = 4 1 = 4because 1 2 + 1 2 = 1.So 41/2 equals2,as 2 × 2 = 4.Therefore 4 1/2 isthesquarerootof4.Itiswrittenas √ 4andequals2:Similarly,4 1/2 = √ 4 = 2 .9 1/2 = √ 9 = 3 .Andingeneral,anynumber araisedtothepower 1 2a 1/2 = √ a .equalsthesquarerootof a:Sothepower,orindex,associatedwithsquarerootsis 1 2 .Also,inthesamewaythattheindex 1 2representsthesquareroot,otherfractionscanbeusedtorepresentotherroots.Thecuberootofthenumber 4iswrittenas4 1/3 = 3√ 4where 1 istheindexrepresentingcuberoot. Similarly,thefourthrootof5maybewrittenas35 1/4 = 4√ 5,andsoon.The n-throotisrepresentedbytheindex 1/n,andthe n-throotof aiswrittenasa 1/n = n√ a .So,forexample,ifwehave 3√ 64thenthisequals64tothepower 1 3 ;andthen3√64 = 641/3= (4 × 4 × 4) 1/3= 4 .Therearesomeimportantpointsaboutroots,orfractionalpowers,thatweneedtoremember.First,wecanwrite4 1/2 × 4 1/2 = 4√4 ×√4 = 4( √ 4) 2 = 4sothatthesquarerootof4,squared,givesyou4backagain. Infactthesquarerootofanynumber,squared,givesyouthatnumberbackagain.Next,ifwehaveaverysimplequadraticequationtosolve,suchas x 2 = 4,thenthesolutionsare x = +2or x = −2.Therearetworoots,as (+2) × (+2) = 4andalso (−2) × (−2) = 4.Wecanwritetherootsas ±2. Sonotallrootsareunique. Butinalotofcircumstancesweonlyneedthepositiveroot,andyoudonothavetoputaplussigninfrontofthesquarerootforthepositiveroot.Byconvention,ifthereisnosigninfrontofthesquarerootthentherootistakentobepositive.Ontheotherhand,supposeweweregiven √ −9.Couldweworkthisoutandgetarealanswer?Now √−9 = (−9) 1/2 ,andsowearelookingforanumberwhichmultipliedbyitselfgives −9. Butthereisnosuchnumber,because 3 × 3 = 9andalso (−3) × (−3) = 9.Soyoucannotfindthesquarerootofanegativenumberandgetarealanswer.3 mc-TY-surds-2009-1 www.mathcentre.ac.uk c○mathcentreJune23,2009
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