{"id":13090,"date":"2014-01-27T14:23:55","date_gmt":"2014-01-27T16:23:55","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=13090"},"modified":"2023-01-13T02:47:44","modified_gmt":"2023-01-13T05:47:44","slug":"problema-olimpico-nivel-b-simplificando-potencias","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-olimpico-nivel-b-simplificando-potencias\/","title":{"rendered":".Problema Ol\u00edmpico – N\u00edvel B: Simplificando pot\u00eancias"},"content":{"rendered":"
\n

Problema<\/strong><\/p>\n

\nDeterminar o valor de
\n [tex]\\qquad \\qquad \\quad \\dfrac{2^n\\times10^{n+2}}{5^{n+1}\\times4^n}[\/tex].<\/div>\n

 <\/p>\n

\n

Solu\u00e7\u00e3o<\/strong><\/p>\n

\nTemos que determinar o valor de [tex]\\dfrac{2^n\\times10^{n+2}}{5^{n+1}\\times4^n}[\/tex].
\nDas propriedades da potencia\u00e7\u00e3o, tem-se que [tex]\\fcolorbox{black}{yellow}{$ \\, a^n\\times a^m=a^{n+m} \\, $}\\,[\/tex]; ent\u00e3o, podemos escrever:
\n 
\n[tex]\\qquad \\qquad \\dfrac{2^n\\times10^{n+2}}{4^n\\times5^{n+1}}=\\dfrac{2^n\\times10^n\\times10^2}{4^n\\times5^n\\times5}=\\dfrac{10^2}{5}\\,\\dfrac{2^n\\times10^n}{4^n\\times5^n}[\/tex].
\n 
\nDe outra propriedade da potencia\u00e7\u00e3o, segue que [tex]\\fcolorbox{black}{yellow}{$ \\, a^n\\times b^n=(a\\times b)^n \\, $}\\,[\/tex]; assim, podemos escrever:
\n 
\n[tex]\\qquad \\qquad \\dfrac{10^2}{5}\\dfrac{2^n\\times10^n}{4^n\\times5^n}=\\dfrac{100}{5}\\,\\dfrac{(2\\times10)^n}{(4\\times5)^n}=20\\dfrac{20^n}{20^n}=20[\/tex].
\n 
\nPortanto,
\n[tex]\\qquad \\qquad \\fbox{$ \\, \\dfrac{2^n\\times10^{n+2}}{5^{n+1}\\times4^n}=20 \\, $}[\/tex].<\/p>\n
\n

Solu\u00e7\u00e3o elaborada pelo aluno do PIC-OBMEP Andr\u00e9 Oliveira Mendes<\/strong>, com contribui\u00e7\u00f5es dos Moderadores do Blog. <\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Problema Determinar o valor de [tex]\\qquad \\qquad \\quad \\dfrac{2^n\\times10^{n+2}}{5^{n+1}\\times4^n}[\/tex].   Solu\u00e7\u00e3o Temos que determinar o valor de [tex]\\dfrac{2^n\\times10^{n+2}}{5^{n+1}\\times4^n}[\/tex]. Das propriedades da potencia\u00e7\u00e3o, tem-se que [tex]\\fcolorbox{black}{yellow}{$ \\, a^n\\times a^m=a^{n+m} \\, $}\\,[\/tex]; ent\u00e3o, podemos escrever:   [tex]\\qquad \\qquad \\dfrac{2^n\\times10^{n+2}}{4^n\\times5^{n+1}}=\\dfrac{2^n\\times10^n\\times10^2}{4^n\\times5^n\\times5}=\\dfrac{10^2}{5}\\,\\dfrac{2^n\\times10^n}{4^n\\times5^n}[\/tex].   De outra propriedade da potencia\u00e7\u00e3o, segue que [tex]\\fcolorbox{black}{yellow}{$ \\, a^n\\times b^n=(a\\times b)^n \\, $}\\,[\/tex]; assim, podemos escrever: … <\/p>\n

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