{"id":99041,"date":"2018-03-02T20:04:44","date_gmt":"2018-03-02T23:04:44","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=99041"},"modified":"2023-04-15T20:31:49","modified_gmt":"2023-04-15T23:31:49","slug":"problema-verdadeiro-ou-falso-2","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-verdadeiro-ou-falso-2\/","title":{"rendered":".Problema: Verdadeiro ou Falso (2)"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:45px;\">\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\"\/>\nAnalise as proposi\u00e7\u00f5es abaixo e classifique-as como verdadeiras ou  falsas. Justifique suas respostas.<br \/>\n [tex]\\textcolor{#800000}{(a)}[\/tex]  [tex]\\sqrt{\\left(2^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}\\times \\sqrt{3\\sqrt{\\left(3^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}}[\/tex] \u00e9 irracional.<\/p>\n<p style=\"margin:10px\">\n<p> [tex]\\textcolor{#800000}{(b)}[\/tex] O n\u00famero [tex]\\boxed{0,7811^2-0,2189^2}[\/tex] \u00e9 maior do que [tex]\\dfrac{4}{7}.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p> [tex]\\textcolor{#800000}{(c)}[\/tex] Se [tex]x=\\dfrac{100000\\times(0,00001)^2\\times 0,001}{0,0001}[\/tex], ent\u00e3o [tex]x=\\dfrac{1}{10}.[\/tex]<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\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[tex]\\textcolor{#800000}{(a)}[\/tex] <strong>Afirma\u00e7\u00e3o falsa<\/strong>.<br \/>\nObserve que:<br \/>\n[tex]\\begin{align*}\\sqrt{\\left(2^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}\\times \\sqrt{3\\sqrt{\\left(3^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}}&#038;=\\sqrt{2^{{\\left(\\sqrt{3}+1\\right)}\\times{\\left(\\sqrt{3}-1\\right)}}}\\times \\sqrt{3\\sqrt{3^{{\\left(\\sqrt{3}+1\\right)}\\times {\\left(\\sqrt{3}-1\\right)}}}} \\\\<br \/>\n&#038;=\\sqrt{2^{(\\sqrt{3})^2-1^2}}\\times \\sqrt{3\\sqrt{3^{(\\sqrt{3})^2-1^2}}}\\\\<br \/>\n&#038;=\\sqrt{2^{3-1}}\\times \\sqrt{3\\sqrt{3^{3-1}}}\\\\<br \/>\n&#038;=\\sqrt{2^{2}}\\times \\sqrt{3\\sqrt{3^{2}}} \\\\<br \/>\n&#038;=2\\times \\sqrt{3\\times 3}\\\\<br \/>\n&#038;=2\\times 3 =6.\\end{align*}[\/tex]<br \/>\nLogo, o n\u00famero em quest\u00e3o \u00e9 racional.<\/p>\n<p style=\"margin:20px\">\n<p>[tex]\\textcolor{#800000}{(b)}[\/tex] <strong>Afirma\u00e7\u00e3o falsa<\/strong>.<br \/>\n Temos que<\/p>\n<ul>\n<li>[tex] \\begin{align*}0,7811^2-0,2189^2&#038;=(0,7811+0,2189)\\times (0,7811-0,2189)\\\\<br \/>\n&#038;=1\\times 0,5622=0,5622 \\end{align*}[\/tex]<\/li>\n<\/ul>\n<p>e que <\/p>\n<ul>\n<li> [tex] \\dfrac{4}{7}=0,571428 \\cdots \\, .[\/tex]<\/li>\n<\/ul>\n<p>Logo<\/p>\n<ul>\n<li>[tex] 0,7811^2-0,2189^2=0,5622  \\lt 0,57 \\lt 0,571428 \\cdots= \\dfrac{4}{7}[\/tex]<\/li>\n<\/ul>\n<p>e, portanto, [tex]\\boxed{0,7811^2-0,2189^2 \\lt \\frac{4}{7}} \\, .[\/tex]<\/p>\n<p style=\"margin:20px\">\n<p>[tex]\\textcolor{#800000}{(c)}[\/tex] <strong>Afirma\u00e7\u00e3o falsa<\/strong>.<br \/>\nObserve:<br \/>\n[tex]\\qquad x=\\dfrac{100000\\times (0,00001)^2\\times 0,001}{0,0001}\\\\[\/tex]<br \/>\n[tex]\\qquad x=\\dfrac{10^5\\times (10^{-5})^2\\times 10^{-3}}{10^{-4}}\\\\[\/tex]<br \/>\n[tex]\\qquad x=\\dfrac{10^5\\times (10^{-10})\\times 10^{-3}}{10^{-4}}\\\\[\/tex]<br \/>\n[tex]\\qquad x=\\dfrac{10^{5-10-3}}{10^{-4}}\\\\[\/tex]<br \/>\n[tex]\\qquad x= \\dfrac{10^{-8}}{10^{-4}}\\\\[\/tex]<br \/>\n[tex]\\qquad x=10^{-8+4}=10^{-4}\\\\[\/tex]<br \/>\n[tex]\\qquad x=\\dfrac{1}{10000}.[\/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:30px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:530px;\">\n<tr>\n<td style=\"border: 1px solid #ffffff\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"95\" height=\"115\" \/><\/td>\n<td  style=\"border: 1px solid  #ffffff\">\n<p style=\"text-align: center;padding: 10px; border-style: dotted;border-color :#800000; border-width :thin; width:400px; background-color: #eee0e5;\"><span style=\"color: #1C1C1C; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px;\">  Para estudar algumas das propriedades utilizadas aqui, e outras mais, visite  clique <a href=\" http:\/\/clubes.obmep.org.br\/blog\/?p=67064\" target=\"_blank\"><strong>esta p\u00e1gina<\/strong><\/a>.<\/span><\/p>\n<p><\/td>\n<\/tr>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 9\u00ba ano do E. F.) Analise as proposi\u00e7\u00f5es abaixo e classifique-as como verdadeiras ou falsas. Justifique suas respostas. [tex]\\textcolor{#800000}{(a)}[\/tex] [tex]\\sqrt{\\left(2^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}\\times \\sqrt{3\\sqrt{\\left(3^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}}[\/tex] \u00e9 irracional. [tex]\\textcolor{#800000}{(b)}[\/tex] O n\u00famero [tex]\\boxed{0,7811^2-0,2189^2}[\/tex] \u00e9 maior do que [tex]\\dfrac{4}{7}.[\/tex] [tex]\\textcolor{#800000}{(c)}[\/tex] Se [tex]x=\\dfrac{100000\\times(0,00001)^2\\times 0,001}{0,0001}[\/tex], ent\u00e3o [tex]x=\\dfrac{1}{10}.[\/tex] Solu\u00e7\u00e3o [tex]\\textcolor{#800000}{(a)}[\/tex] Afirma\u00e7\u00e3o falsa. Observe que: [tex]\\begin{align*}\\sqrt{\\left(2^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}\\times \\sqrt{3\\sqrt{\\left(3^{\\sqrt{3}+1}\\right)^{\\sqrt{3}-1}}}&#038;=\\sqrt{2^{{\\left(\\sqrt{3}+1\\right)}\\times{\\left(\\sqrt{3}-1\\right)}}}\\times \\sqrt{3\\sqrt{3^{{\\left(\\sqrt{3}+1\\right)}\\times {\\left(\\sqrt{3}-1\\right)}}}} \\\\ &#038;=\\sqrt{2^{(\\sqrt{3})^2-1^2}}\\times &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-verdadeiro-ou-falso-2\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":620,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-99041","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/99041","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\/620"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=99041"}],"version-history":[{"count":12,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/99041\/revisions"}],"predecessor-version":[{"id":187243,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/99041\/revisions\/187243"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=99041"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}