Surds, and other roots - Math Centre
Surds, and other roots - Math Centre
Surds, and other roots - Math Centre
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Key PointThesquarerootof aiswrittenas a 1/2 <strong>and</strong>isequalto √ a.The n-throotof aiswrittenas a 1/n<strong>and</strong>isequalto n√ a.Theformula√ a ×√ a = acanbeusedtosimplifyexpressionsinvolvingsquare<strong>roots</strong>.Notall<strong>roots</strong>areunique,forexamplethesquarerootof4is2or −2,sometimeswrittenas±2.Butwhenwrittenwithoutasigninfront,thesquarerootrepresentsthepositiveroot.Youcannotfindthesquarerootofanegativenumber.3. <strong>Surds</strong> <strong>and</strong> irrational numbersWeshallnowlookatsomesquare<strong>roots</strong>inmoredetail. Take,forexample, √ √25:itsvalueis5.9Andthevalueof is 3 ,or 4 2 11. Sosomesquare<strong>roots</strong>canbeevaluatedaswholenumbersor2asfractions,in<strong>other</strong>wordsasrationalnumbers.Butwhatabout √ 2or √ 3?The<strong>roots</strong>tothesearenotwholenumbersorfractions,<strong>and</strong>sotheyhaveirrationalvalues.Theyareusuallywrittenasdecimalstoagivenapproximation.Forexample√2 = 1.414√3 = 1.732to3decimalplaces,to3decimalplaces.Whenwehavesquare<strong>roots</strong>whichgiveirrationalnumberswecallthemsurds. So √ 2<strong>and</strong> √ 3aresurds.Othersurdsare√5,√6,√7,√8,√10 <strong>and</strong>soon.<strong>Surds</strong>areoftenfoundwhenusingPythagoras’Theorem,<strong>and</strong>intrigonometry.So,wherepossible,itisusefultobeabletosimplifyexpressionsinvolvingsurds. Take,forexample, √ 8. Thiscanbewrittenas √ 4 × 2,whichwecanrewriteas √ 4 × √ 2,in<strong>other</strong>wordsas 2 √ 2:√8 =√4 × 2= √ 4 × √ 2= 2 √ 2.Ingeneral,thesquarerootofaproductistheproductofthesquare<strong>roots</strong>,<strong>and</strong>viceversa.Thisisusefultoknowwhensimplifyingsurdexpressions.c○mathcentreJune23,2009 www.mathcentre.ac.uk 4 mc-TY-surds-2009-1
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