{"id":8950,"date":"2012-05-06T21:51:25","date_gmt":"2012-05-06T20:51:25","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8950"},"modified":"2021-12-29T00:56:19","modified_gmt":"2021-12-29T00:56:19","slug":"mostre-que-5","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8950","title":{"rendered":"Mostre que"},"content":{"rendered":"<p><ul id='GTTabs_ul_8950' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8950' class='GTTabs_curr'><a  id=\"8950_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8950' ><a  id=\"8950_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_8950'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Mostre, pela defini\u00e7\u00e3o, que $$\\frac{{\\sqrt 2 }}{2} &#8211; \\frac{{\\sqrt 2 }}{2}i$$ \u00e9 uma das ra\u00edzes quartas de $-1$.<\/p>\n<p>(Recorra ao Bin\u00f3mio de Newton)<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8950' onClick='GTTabs_show(1,8950)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8950'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Recorrendo ao Bin\u00f3mio de Newton, vem:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {\\frac{{\\sqrt 2 }}{2} &#8211; \\frac{{\\sqrt 2 }}{2}i} \\right)}^4}}&amp; = &amp;{\\sum\\limits_{k = 0}^4 {{}^4{C_k} \\times {{\\left( {\\frac{{\\sqrt 2 }}{2}} \\right)}^{4 &#8211; k}} \\times {{\\left( { &#8211; \\frac{{\\sqrt 2 }}{2}i} \\right)}^k}} } \\\\<br \/>\n{}&amp; = &amp;{1 \\times \\frac{4}{{16}} \\times 1 + 4 \\times \\frac{{2\\sqrt 2 }}{8} \\times \\left( { &#8211; \\frac{{\\sqrt 2 }}{2}i} \\right) + 6 \\times \\frac{2}{4} \\times \\left( { &#8211; \\frac{2}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{1}{4} &#8211; i &#8211; \\frac{3}{2} + i + \\frac{1}{4}} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 1}<br \/>\n\\end{array} + 4 \\times \\frac{{\\sqrt 2 }}{2} \\times \\frac{{2\\sqrt 2 }}{8}i + 1 \\times 1 \\times \\frac{4}{{16}}$$<\/p>\n<p>Como ${\\left( {\\frac{{\\sqrt 2 }}{2} &#8211; \\frac{{\\sqrt 2 }}{2}i} \\right)^4} =\u00a0 &#8211; 1$, ent\u00e3o $\\frac{{\\sqrt 2 }}{2} &#8211; \\frac{{\\sqrt 2 }}{2}i$ \u00e9 uma das ra\u00edzes quartas de $-1$.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8950' onClick='GTTabs_show(0,8950)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Mostre, pela defini\u00e7\u00e3o, que $$\\frac{{\\sqrt 2 }}{2} &#8211; \\frac{{\\sqrt 2 }}{2}i$$ \u00e9 uma das ra\u00edzes quartas de $-1$. (Recorra ao Bin\u00f3mio de Newton) Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19524,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18],"series":[],"class_list":["post-8950","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-numeros-complexos"],"views":1886,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Binomio_de_Newton.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8950"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8950\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19524"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8950"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8950"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8950"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}