{"id":36249,"date":"2015-05-03T16:47:22","date_gmt":"2015-05-03T19:47:22","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=36249"},"modified":"2023-03-21T22:02:35","modified_gmt":"2023-03-22T01:02:35","slug":"desafio-texalpha-beta-tex","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/desafio-texalpha-beta-tex\/","title":{"rendered":".Desafio: [tex]\\alpha = \\beta ?[\/tex]"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nConsidere os n\u00fameros <\/p>\n<p>[tex]\\qquad \\alpha=\\sqrt{13}+\\sqrt{10+2\\sqrt{13}}[\/tex]<br \/>\ne<br \/>\n [tex]\\qquad \\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{65-26\\sqrt{3}}}[\/tex] .<\/p>\n<p>Mostre que [tex]\\alpha=\\beta[\/tex].\n<\/p><\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\">\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<ul>\n<li> Inicialmente, observe que\n<p>[tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{65-26\\sqrt{3}}}[\/tex]<\/p>\n<p>[tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{13}\\sqrt{5-2\\sqrt{3}}}[\/tex]<\/p>\n<p>[tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{\\left(\\sqrt{13}+\\sqrt{5-2\\sqrt{3}}\\right)^2}[\/tex]<\/p>\n<p>[tex]\\qquad\\qquad\\boxed{\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{13}+\\sqrt{5-2\\sqrt{3}}}[\/tex].<\/li>\n<p style=\"margin:30px\">\n<li>Agora, seja [tex]\\boxed{x=\\sqrt{5+2\\sqrt{3}}+\\sqrt{5-2\\sqrt{3}}}[\/tex].<br \/>\nAssim:<br \/>\n[tex]\\qquad\\qquad x^2=5+2\\sqrt{3}+2\\sqrt{5+2\\sqrt{3}}\\sqrt{5-2\\sqrt{3}}+5-2\\sqrt{3}[\/tex]<\/p>\n<p>[tex]\\qquad\\qquad x^2=10+2\\sqrt{25-12} [\/tex]<\/p>\n<p>[tex]\\qquad\\qquad x^2=10+2\\sqrt{13} [\/tex]<\/p>\n<p>[tex]\\qquad\\qquad x=\\pm\\sqrt{10+2\\sqrt{13}}. [\/tex]<\/p>\n<p>Como [tex]x\\gt 0[\/tex], ent\u00e3o [tex] \\boxed{x=\\sqrt{10+2\\sqrt{13}}}[\/tex].<\/li>\n<\/ul>\n<p>Portanto, pelo exposto, <\/p>\n<p>[tex]\\qquad \\qquad  \\alpha=\\sqrt{13}+\\sqrt{10+2\\sqrt{13}}[\/tex] <\/p>\n<p>[tex]\\qquad \\qquad  \\alpha=\\sqrt{13}+x[\/tex] <\/p>\n<p>[tex]\\qquad \\qquad  \\alpha=\\sqrt{13}+\\left[\\sqrt{5+2\\sqrt{3}}+\\sqrt{5-2\\sqrt{3}}\\right][\/tex] <\/p>\n<p>[tex]\\qquad \\qquad  \\alpha=\\sqrt{5+2\\sqrt{3}}+\\sqrt{13}+\\sqrt{5-2\\sqrt{3}}[\/tex] <\/p>\n<p>[tex]\\qquad\\qquad\\alpha=\\beta[\/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<p style=\"margin:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 9\u00ba ano do E. F.) Considere os n\u00fameros [tex]\\qquad \\alpha=\\sqrt{13}+\\sqrt{10+2\\sqrt{13}}[\/tex] e [tex]\\qquad \\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{65-26\\sqrt{3}}}[\/tex] . Mostre que [tex]\\alpha=\\beta[\/tex]. Solu\u00e7\u00e3o Inicialmente, observe que [tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{65-26\\sqrt{3}}}[\/tex] [tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{13}\\sqrt{5-2\\sqrt{3}}}[\/tex] [tex]\\qquad\\qquad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{\\left(\\sqrt{13}+\\sqrt{5-2\\sqrt{3}}\\right)^2}[\/tex] [tex]\\qquad\\qquad\\boxed{\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{13}+\\sqrt{5-2\\sqrt{3}}}[\/tex]. Agora, seja [tex]\\boxed{x=\\sqrt{5+2\\sqrt{3}}+\\sqrt{5-2\\sqrt{3}}}[\/tex]. Assim: [tex]\\qquad\\qquad x^2=5+2\\sqrt{3}+2\\sqrt{5+2\\sqrt{3}}\\sqrt{5-2\\sqrt{3}}+5-2\\sqrt{3}[\/tex] [tex]\\qquad\\qquad x^2=10+2\\sqrt{25-12} [\/tex] [tex]\\qquad\\qquad x^2=10+2\\sqrt{13} [\/tex] [tex]\\qquad\\qquad x=\\pm\\sqrt{10+2\\sqrt{13}}. [\/tex] Como [tex]x\\gt 0[\/tex], ent\u00e3o [tex] \\boxed{x=\\sqrt{10+2\\sqrt{13}}}[\/tex]. Portanto, pelo exposto, [tex]\\qquad &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/desafio-texalpha-beta-tex\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-36249","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36249","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=36249"}],"version-history":[{"count":37,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36249\/revisions"}],"predecessor-version":[{"id":185090,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36249\/revisions\/185090"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=36249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}