mathspassion Posted January 6, 2017 Share Posted January 6, 2017 (edited) Edited January 6, 2017 by mathspassion Link to comment Share on other sites More sharing options...
imatfaal Posted January 6, 2017 Share Posted January 6, 2017 Again - similar to your last "discovery" - this is a simple artifact of the number system and multiplication. If you look at the basis of Pascal and Binomial expansion for (a+x)^6 to extend you get as follows [latex]1a^6x^0+6a^5x^1+15a^4x^2+20a^3x^3+15a^2x^4+6a^1x^5+1a^0x^6[/latex] If you set a to be 10 and x to be 1 (ie a is the tens and x is the units in decimal counting) this simplifies to [latex]1\cdot10^6\cdot1+6\cdot10^5\cdot1+15\cdot10^4\cdot1+20\cdot10^3\cdot1+15\cdot10^2\cdot1+6\cdot10^1\cdot1+1\cdot10^0\cdot1[/latex] simplify again [latex]1\cdot10^6+6\cdot10^5+15\cdot10^4+20\cdot10^3+15\cdot10^2+6\cdot10^1+1[/latex] and again 1,000,000 + 600,000 + 150,000 + 20,000 + 1,500 + 60 + 1 = 1,771,561 = 11^6 It works with any number and any power - it is just more obvious when both a and x are one 1 Link to comment Share on other sites More sharing options...
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