Rules Of Indices

Introduction
math2ever

Four important rules of indices are:

  • Rule of Multiplication
  • Rule of Division
  • Rule of Power of a Power
  • Rule of Powers of Brackets





Four important rules of indices are listed in table 1. Rules of multiplication and division are applied for indices containing identical base.

No.        Rules of indicesExamples
1Rule of Multiplication
a m × a n = a m+n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fcaWGHbWaaWbaaSqabeaacaWGTbaaaOGaey41aqRaamyyamaaCaaaleqabaGaamOBaaaakiabg2da9iaadggadaahaaWcbeqaaiaad2gacqGHRaWkcaWGUbaaaaaa@4698@
2 3 × 2 4 = 2 3+4 = 2 7 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabyaauqaaaBaaaWaa82aa85aauEraaiaaikdadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaIYaWaaWbaaSqabeaacaaI0aaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaG4maiabgUcaRiaaisdaaaaakeaacaWLjaGaeyypa0JaaGOmamaaCaaaleqabaGaaG4naaaaaaaa@48C1@
2Rule of Division
a m ÷ a n = a mn MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fcaWGHbWaaWbaaSqabeaacaWGTbaaaOGaey49aGRaamyyamaaCaaaleqabaGaamOBaaaakiabg2da9iaadggadaahaaWcbeqaaiaad2gacqGHsislcaWGUbaaaaaa@46C7@
3 4 ÷ 3 2 = 3 42 = 3 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabyaaeraaaBaaaWaa82aa85aauEraaiaaiodadaahaaWcbeqaaiaaisdaaaGccqGH3daUcaaIZaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaG4mamaaCaaaleqabaGaaGinaiabgkHiTiaaikdaaaaakeaacaWLjaGaeyypa0JaaG4mamaaCaaaleqabaGaaGOmaaaaaaaa@491D@
3Rule for power of a power
( a m ) n = a m×n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fdaqadaqacuaaaBaaaWaa82aa85aauEHaamyyamaaCaaaleqabaGaamyBaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaakiabg2da9iaadggadaahaaWcbeqaaiaad2gacqGHxdaTcaWGUbaaaaaa@4930@
( 4 2 ) 3 = 4 2×3 = 4 6 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabyaaeoaaaBaaaWaa82aa85aauEraamaabmaabiqbaaSbaaadaWBdaWNdaq5fcaaI0aWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGinamaaCaaaleqabaGaaGOmaiabgEna0kaaiodaaaaakeaacaWLjaGaeyypa0JaaGinamaaCaaaleqabaGaaGOnaaaaaaaa@4BF3@
4Rule of Powers of Brackets
( a m b n c p ) r = a m×r b n×r c p×r MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fdaqadaqacuaaaBaaaWaa82aa85aauEXaaSaaaeGafaaWgaaamaaVnaaFoaaLxiaadggadaahaaWcbeqaaiaad2gaaaGccaWGIbWaaWbaaSqabeaacaWGUbaaaaGcbaGaam4yamaaCaaaleqabaGaamiCaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadkhaaaGccqGH9aqpdaWcaaqacuaaaBaaaWaa82aa85aauEHaamyyamaaCaaaleqabaGaamyBaiabgEna0kaadkhaaaGccaWGIbWaaWbaaSqabeaacaWGUbGaey41aqRaamOCaaaaaOqaaiaadogadaahaaWcbeqaaiaadchacqGHxdaTcaWGYbaaaaaaaaa@5D6B@
( 3ab 2 c 2 ) 2 = 3 2 a 2 b 2 2 2 c 2×2 = 9 a 2 b 2 4 c 4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabeaa4tqaamaabmaabaWaaSaaaeaacaaIZaGaamyyaiaadkgaaeaacaaIYaGaam4yamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaaiodadaahaaWcbeqaaiaaikdaaaGccaWGHbWaaWbaaSqabeaacaaIYaaaaOGaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdadaahaaWcbeqaaiaaikdaaaGccaWGJbWaaWbaaSqabeaacaaIYaGaey41aqRaaGOmaaaaaaaakeaacaWLjaGaeyypa0ZaaSaaaeaacaaI5aGaamyyamaaCaaaleqabaGaaGOmaaaakiaadkgadaahaaWcbeqaaiaaikdaaaaakeaacaaI0aGaam4yamaaCaaaleqabaGaaGinaaaaaaaaaaa@56F2@

Other important rules - applied for indices containing different base.
No.        Rules of indicesExamples
1Different base but similar power
a m × b m = ( a×b ) m c n ÷ d n = ( c d ) n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabuaaaBaaaWaa82aa85aauEraaiaadggadaahaaWcbeqaaiaad2gaaaGccqGHxdaTcaWGIbWaaWbaaSqabeaacaWGTbaaaOGaeyypa0ZaaeWaaeaacaWGHbGaey41aqRaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaamyBaaaaaOqaaiaadogadaahaaWcbeqaaiaad6gaaaGccqGH3daUcaWGKbWaaWbaaSqabeaacaWGUbaaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaadogaaeaacaWGKbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaaaaaa@554A@
e 3 × f 3 = ( e×f ) 3 = ( ef ) 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGceiqabyaaqraaaBaaaWaa82aa85aauEraaiaadwgadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaWGMbWaaWbaaSqabeaacaaIZaaaaOGaeyypa0ZaaeWaaeaacaWGLbGaey41aqRaamOzaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaOqaaiaaxMaacqGH9aqpdaqadaqaaiaadwgacaWGMbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaa@4F10@
2Zero Power
a 0 =1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fcaWGHbWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGymaaaa@3F19@
6 0 =1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiGbaqebaaSbaaadaWBdaWNdaq5fcaaI2aWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGymaaaa@3F3C@
3Negative Power
1 a m = a m MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fdaWcaaqacuaaaBaaaWaa82aa85aauEHaaGymaaqacuaaaBaaaWaa82aa85aauEHaamyyamaaCaaaleqabaGaamyBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacqGHsislcaWGTbaaaaaa@4801@
1 4 2 = 4 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiGbaqebaaSbaaadaWBdaWNdaq5fdaWcaaqaaiaaigdaaeaacaaI0aWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9iaaisdadaahaaWcbeqaaiabgkHiTiaaikdaaaaaaa@41E0@
4Rule of Root
a n m = a n m MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fdaGcbaqaaiaadggadaahaaWcbeqaaiaad6gaaaaabaGaamyBaaaakiabg2da9iaadggadaahaaWcbeqaamaalaaabaGaamOBaaqaaiaad2gaaaaaaaaa@42A1@
a 2 3 = a 2 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIjxAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGabiGaaiaabeqaamaaeaqbaaGcbiqbaaSbaaadaWBdaWNdaq5fdaGcbaqaaiaadggadaahaaWcbeqaaiaaikdaaaaabaGaaG4maaaakiabg2da9iaadggadaahaaWcbeqaamaalaaabaGaaGOmaaqaaiaaiodaaaaaaaaa@41C9@