{"id":101926,"date":"2018-04-23T20:11:34","date_gmt":"2018-04-23T23:11:34","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=101926"},"modified":"2023-04-19T18:09:30","modified_gmt":"2023-04-19T21:09:30","slug":"problemao-determine-o-valor-de-x","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-determine-o-valor-de-x\/","title":{"rendered":".Problem\u00e3o: Determine o valor de x"},"content":{"rendered":"<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:40px;\">\n<p class=\"tit\" style=\"line-height:20px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 2\u00ba ano do E. M.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Sabendo que<br \/>\n[tex]\\qquad \\qquad x=\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}}} \\, [\/tex].<br \/>\n\u00e9 um n\u00famero real positivo, determine seu valor.<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:40px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Podemos reescrever [tex]x[\/tex] como:<br \/>\n[tex]\\qquad \\qquad x=\\left(27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}}\\right)^{\\frac{1}{4}} \\, [\/tex]<br \/>\n[tex]\\qquad \\qquad x=\\left(27\\left(27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex]<br \/>\n[tex]\\qquad \\qquad x=\\left(27\\left(27\\left(27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex]<br \/>\n[tex]\\qquad \\qquad x=\\left(27\\left(27\\left(27\\left(27\\sqrt[4]{27\\cdots}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex]<br \/>\n[tex]\\qquad \\qquad x=27^{\\frac{1}{4}}\\times 27^{\\frac{1}{4}\\times \\frac{1}{4}}\\times 27^{\\frac{1}{4}\\times \\frac{1}{4}\\times \\frac{1}{4}}\\times 27^{\\frac{1}{4}\\times \\frac{1}{4}\\times \\frac{1}{4}\\times \\frac{1}{4}}\\times \\cdots \\, [\/tex].<br \/>\nAssim, temos que:<br \/>\n[tex]\\qquad \\qquad x=27^{\\frac{1}{4}+\\left(\\frac{1}{4}\\right)^2+\\left(\\frac{1}{4}\\right)^3+\\left(\\frac{1}{4}\\right)^4+\\cdots} \\, .\\qquad \\textcolor{#800000}{(i)}[\/tex]<br \/>\nObserve que o expoente \u00e9 a soma de infinitos termos de uma Progress\u00e3o Geom\u00e9trica de raz\u00e3o [tex]\\frac{1}{4}[\/tex]; logo, a P.G.  converge e a soma dos seus termos \u00e9 dada por:<br \/>\n[tex]\\qquad \\qquad \\boxed{\\dfrac{1}{4}+\\left(\\dfrac{1}{4}\\right)^2+\\left(\\dfrac{1}{4}\\right)^3+\\left(\\dfrac{1}{4}\\right)^4+\\cdots}=\\dfrac{\\frac{1}{4}}{1-\\frac{1}{4}}=\\dfrac{\\frac{1}{4}}{\\frac{3}{4}}=\\dfrac{1}{4}\\cdot \\dfrac{4}{3}=\\boxed{\\dfrac{1}{3}} \\, .[\/tex]<br \/>\nEnt\u00e3o, de [tex]\\textcolor{#800000}{(i)}[\/tex], segue que:<br \/>\n[tex]\\qquad \\qquad x=27^{\\frac{1}{3}}=(3^3)^\\frac{1}{3}=3^{\\left(3\\times\\frac{1}{3}\\right)}=3^1=3[\/tex].<br \/>\nPortanto, [tex]\\fcolorbox{black}{#eee0e5}{$x=3$} \\, .[\/tex]<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 2\u00ba ano do E. M.) Sabendo que [tex]\\qquad \\qquad x=\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}}} \\, [\/tex]. \u00e9 um n\u00famero real positivo, determine seu valor. Solu\u00e7\u00e3o Podemos reescrever [tex]x[\/tex] como: [tex]\\qquad \\qquad x=\\left(27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}}\\right)^{\\frac{1}{4}} \\, [\/tex] [tex]\\qquad \\qquad x=\\left(27\\left(27\\sqrt[4]{27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex] [tex]\\qquad \\qquad x=\\left(27\\left(27\\left(27\\sqrt[4]{27\\sqrt[4]{27\\cdots}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex] [tex]\\qquad \\qquad x=\\left(27\\left(27\\left(27\\left(27\\sqrt[4]{27\\cdots}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}}\\right)^{\\frac{1}{4}} \\, [\/tex] [tex]\\qquad \\qquad x=27^{\\frac{1}{4}}\\times 27^{\\frac{1}{4}\\times \\frac{1}{4}}\\times &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-determine-o-valor-de-x\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":667,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-101926","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/101926","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/667"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=101926"}],"version-history":[{"count":7,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/101926\/revisions"}],"predecessor-version":[{"id":187608,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/101926\/revisions\/187608"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=101926"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}