Свойства степени с натуральным показателем 7 класс

1 свойство

При умножении разных степеней с одинаковыми основаниями основание степеней оставляется

прежним, а показатели складываются:

a m      a n  =  a m  +  n .

a 3      a 4  = a ⋅ a ⋅ a ⏟ 3 раза size 12{ { size 11{a cdot a cdot a}} underbrace { size 8{3``"раза"} } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAX4EAAD/ AwAAQAAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAH0EAAD+AwAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAAgwEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBhwQAA AAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AAAAAACDAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAACQEAAI0BAAABAAAAxSLfAAAAAAD//wEA xSKMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAAgwEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAMIBAACNAQAAAQBhwQAAAAAA//8BAGEA jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACDAQAA //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD///// AIYAAQAEAAAAAAAAAHIAAgAaAAAAYQIAAI0BAAABAAAAxSLfAAAAAAD//wEAxSKMAAEAAAAA AIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAAgwEA AAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA//// /wCGAAEABAAAAAAAAAByAAIAFwAAABoDAACNAQAAAQBhwQAAAAAA//8BAGEAjAABAAAAAACL AAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAADIDAAB5AgAA//8AAAAABQAA AAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAA AAAAAHIAAgAaAAAAhAAAAPcBAAABAAAA3yNjAwAAAAD//wEA3yOMAAEAAAAAAIsAAQACAAAA HwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAAGQEAAAAAAwAAAAUA AAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAA AAAAAAByAAIAFwAAAJ8AAAC5AwAAAQAzjQAAAAAA//8BADMAjAABAAAAAACLAAEAAgAAAB8A igABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAABkBAAAAAAMAAAAFAAAA AAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA AAAAcgACACQAAACnAQAAuQMAAAgA0YDQsNC30LDsAQAAAAD//wQAQAQwBDcEMASMAAEAAAAA AJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==  a ⋅ a ⋅ a ⋅ a ⏟ 4 раза size 12{ { size 11{a cdot a cdot a cdot a}} underbrace { size 8{4``"раза"} } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAdIFAAD/ AwAATgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAANEFAAD+AwAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAAgwEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBhwQAA AAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AAAAAACDAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAACQEAAI0BAAABAAAAxSLfAAAAAAD//wEA xSKMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAAgwEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAMIBAACNAQAAAQBhwQAAAAAA//8BAGEA jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACDAQAA //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD///// AIYAAQAEAAAAAAAAAHIAAgAaAAAAYQIAAI0BAAABAAAAxSLfAAAAAAD//wEAxSKMAAEAAAAA AIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAAgwEA AAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA//// /wCGAAEABAAAAAAAAAByAAIAFwAAABoDAACNAQAAAQBhwQAAAAAA//8BAGEAjAABAAAAAACL AAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACDAQAA//8AAAAABQAA AAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAA AAAAAHIAAgAaAAAAuQMAAI0BAAABAAAAxSLfAAAAAAD//wEAxSKMAAEAAAAAAIsAAQACAAAA HwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAAgwEAAAAAAwAAAAUA AAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAA AAAAAAByAAIAFwAAAHIEAACNAQAAAQBhwQAAAAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8A igABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAHwEAAB5AgAA//8AAAAABQAAAAAAAAD/AwAA AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAa AAAAagAAAPcBAAABAAAA3yPBBAAAAAD//wEA3yOMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAA AAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAAGQEAAAAAAwAAAAUAAAAAAAAA/wMA AAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIA FwAAAFgBAAC5AwAAAQA0jQAAAAAA//8BADQAjAABAAAAAACLAAEAAgAAAB8AigABAEQAAAAD AD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAABkBAAAAAAMAAAAFAAAAAAAAAP8DAAAA AAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACACQA AABhAgAAuQMAAAgA0YDQsNC30LDsAQAAAAD//wQAQAQwBDcEMASMAAEAAAAAAJUAAQAEAAAA AAAAAJYAAQACAAAACQCMAAEAAAAAAA== = a ⋅ a ⋅ . . . ⋅ a ⏟ 3 + 4 = 7 раз size 12{ { size 11{a cdot a cdot "." "." 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2 свойство

При умножении одинаковых степеней с разными основаниями эти основания перемножаются, а

показатель степени остается прежним:

a n      b n  = ( ab ) n .

a 3      b 3  =  a      a      a      b      b      b  = 

=( a      b )    ( a      b )    ( a      b ) = ( ab ) 3

m , a ≠ 0." width="640"

3 свойство

При делении степени на степень с тем же основанием основание остается прежним, а показатели вычитаются:

a n : a m = a n – m , n m , a ≠ 0.

a 5 a 3 = a ⋅ a ⋅ a ⋅ a ⋅ a a ⋅ a ⋅ a size 12{ { { size 11{a rSup { size 8{5} } }} over { size 12{a rSup { size 8{3} } } } } = { { size 12{a cdot a cdot a cdot a cdot a} } over { size 12{a cdot a cdot a} } } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAATsLAACz BAAAnwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAADoLAACyBAAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAAgwEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAJ8AAADCAQAAAQBhwQAA AAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21h biBObzkgTAAAAAAAABkBAAAAAAMAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQAC AAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAByAQAACQEAAAEANY0AAAAA AP//AQA1AIwAAQAAAAAAiwABAAIAAAAfAIUAAQAFAAAAAAAAAAGEAAEABQAAAAAAAAAAigAB ADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAA AAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAGcAAQAQAAAA hAAAAEYCAAAqAgAAWgIAAIwAAQAAAAAAiwABAAIAAAAfAIoAAQBEAAAAAwA+AAAAEgBOaW1i dXMgUm9tYW4gTm85IEwAAAAAAACmAQAAAAADAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAA AACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAnwAAAD0EAAAB AGHTAAAAAAD//wEAYQCMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVz IFJvbWFuIE5vOSBMAAAAAAAAGQEAAAAAAwAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAA iAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHIBAABpAwAAAQAz jQAAAAAA//8BADMAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1i b2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAARgIAAMoCAAABAAAAPQBPAQAAAAD/ /wEAPQCMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5v OSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAANMDAADCAQAAAQBh0wAAAAAA//8B AGEAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACm AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAaAAAAcgQAAMIBAAABAAAAxSL0AAAAAAD//wEAxSKMAAEA AAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAA pgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA /////wCGAAEABAAAAAAAAAByAAIAFwAAAEUFAADCAQAAAQBh0wAAAAAA//8BAGEAjAABAAAA AACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACmAQAA//8AAAAA BQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE AAAAAAAAAHIAAgAaAAAA5AUAAMIBAAABAAAAxSL0AAAAAAD//wEAxSKMAAEAAAAAAIsAAQAC AAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAApgEAAAAAAwAA AAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEA BAAAAAAAAAByAAIAFwAAALgGAADCAQAAAQBh0wAAAAAA//8BAGEAjAABAAAAAACLAAEAAgAA AB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/ AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA AgAaAAAAcQcAAMIBAAABAAAAxSL0AAAAAAD//wEAxSKMAAEAAAAAAIsAAQACAAAAHwCKAAEA RAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA /wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAABy AAIAFwAAACoIAADCAQAAAQBh0wAAAAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8AigABADwA AAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAA AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAA4wgA AMIBAAABAAAAxSL0AAAAAAD//wEAxSKMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAA ABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAA AAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAJ0J AADCAQAAAQBh0wAAAAAA//8BAGEAjAABAAAAAACLAAEAAgAAAB8AhQABAAUAAAAAAAAAAYQA AQAFAAAAAAAAAACKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKYBAAD//wAAAAAF AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA AAAAAAAAZwABABAAAAC5AwAARgIAAKMKAABaAgAAjAABAAAAAACLAAEAAgAAAB8AigABAEQA AAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8D AAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgAC ABcAAABFBQAA7QMAAAEAYdMAAAAAAP//AQBhAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAA AwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApgEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/ AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAOQFAADt AwAAAQAAAMUi9AAAAAAA//8BAMUijAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAAS AE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA /wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAC4BgAA 7QMAAAEAYdMAAAAAAP//AQBhAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBP cGVuU3ltYm9sAAAAAAAApgEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAHEHAADtAwAAAQAAAMUi 9AAAAAAA//8BAMUijAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBS b21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgA AQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAAqCAAA7QMAAAEAYdMA AAAAAP//AQBhAIwAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA =  a      a  =  a 2

4 свойство

При делении степеней с одинаковыми показателями

основания делятся друг на друга, а показатель степени остается прежним:

a n : b n = , n m , b ≠ 0 .

5 свойство

При возведении степени в степень основание степени остается прежним, а показатели степеней перемножаются:

( a m ) n  =  a m      n

( a 2 ) 3  =  a 2      a 2      a 2  =  a 2 + 2 + 2  =  a 6