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Question Number 130426 by Koyoooo last updated on 25/Jan/21

Answered by Olaf last updated on 25/Jan/21

(9×9×9×9)_(10)  = 100_(81)   9^4  en base 10 egale 100 en base 81

$$\left(\mathrm{9}×\mathrm{9}×\mathrm{9}×\mathrm{9}\right)_{\mathrm{10}} \:=\:\mathrm{100}_{\mathrm{81}} \\ $$$$\mathrm{9}^{\mathrm{4}} \:\mathrm{en}\:\mathrm{base}\:\mathrm{10}\:\mathrm{egale}\:\mathrm{100}\:\mathrm{en}\:\mathrm{base}\:\mathrm{81} \\ $$

Answered by Olaf last updated on 25/Jan/21

(9+9+9+9)_(10)  = 100_6   (9+9+9+9)_(25)  = 100_9   (9+9+9+9)_(46)  = 100_(12)   (9+9+9+9)_(73)  = 100_(15)   (9+9+9+9)_(106)  = 100_(18)   (9+9+9+9)_(145)  = 100_(21)   ...  (100)_(3n+6)  = (9+9+9+9)_(3(n+2)^2 −2)

$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{10}} \:=\:\mathrm{100}_{\mathrm{6}} \\ $$$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{25}} \:=\:\mathrm{100}_{\mathrm{9}} \\ $$$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{46}} \:=\:\mathrm{100}_{\mathrm{12}} \\ $$$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{73}} \:=\:\mathrm{100}_{\mathrm{15}} \\ $$$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{106}} \:=\:\mathrm{100}_{\mathrm{18}} \\ $$$$\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{145}} \:=\:\mathrm{100}_{\mathrm{21}} \\ $$$$... \\ $$$$\left(\mathrm{100}\right)_{\mathrm{3}{n}+\mathrm{6}} \:=\:\left(\mathrm{9}+\mathrm{9}+\mathrm{9}+\mathrm{9}\right)_{\mathrm{3}\left({n}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}} \\ $$

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