{"id":9325,"date":"2012-05-22T21:36:53","date_gmt":"2012-05-22T20:36:53","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9325"},"modified":"2022-01-27T00:32:58","modified_gmt":"2022-01-27T00:32:58","slug":"uma-raiz-cubica-de-um-numero-complexo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9325","title":{"rendered":"Uma raiz c\u00fabica de um n\u00famero complexo"},"content":{"rendered":"<p><ul id='GTTabs_ul_9325' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9325' class='GTTabs_curr'><a  id=\"9325_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9325' ><a  id=\"9325_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_9325'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>${w_1} = \\frac{{ &#8211; 1 + \\sqrt 3 i}}{2}$ \u00e9 uma raiz c\u00fabica de um n\u00famero complexo $z$.<\/p>\n<ol>\n<li>Determine as outras ra\u00edzes c\u00fabicas de $z$.<\/li>\n<li>Determine $z$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9325' onClick='GTTabs_show(1,9325)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9325'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{{w_1}}&amp; = &amp;{\\frac{{ &#8211; 1 + \\sqrt 3 i}}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( {\\frac{{2\\pi }}{3}} \\right)}<br \/>\n\\end{array}$$<\/p>\n<p>As tr\u00eas ra\u00edzes c\u00fabicas possuem igual m\u00f3dulo e os seus argumentos est\u00e3o em progress\u00e3o aritm\u00e9tica de raz\u00e3o $\\frac{{2\\pi }}{3}$.<\/p>\n<p>Logo, as outras duas ra\u00edzes c\u00fabicas s\u00e3o:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{w_2}}&amp; = &amp;{\\operatorname{cis} \\left( {\\frac{{2\\pi }}{3} + \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( {\\frac{{4\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; 1 &#8211; \\sqrt 3 i}}{2}}<br \/>\n\\end{array}$$ e $$\\begin{array}{*{20}{l}}<br \/>\n{{w_3}}&amp; = &amp;{\\operatorname{cis} \\left( {\\frac{{4\\pi }}{3} + \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{cis} \\left( 0 \\right)} \\\\<br \/>\n{}&amp; = &amp;1<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora, $z = {\\left( {{w_1}} \\right)^3} = {\\left( {{w_2}} \\right)^3} = {\\left( {{w_2}} \\right)^3} = 1$.<br \/>\n\u00ad<\/li>\n<\/ol>\n<div id=\"attachment_9330\" style=\"width: 479px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9330\" data-attachment-id=\"9330\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9330\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58.png\" data-orig-size=\"782,755\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"12pag143-58\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58.png\" class=\" wp-image-9330\" title=\"Plano de Argand\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58.png\" alt=\"\" width=\"469\" height=\"453\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58.png 782w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58-300x289.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58-150x144.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-58-400x386.png 400w\" sizes=\"auto, (max-width: 469px) 100vw, 469px\" \/><\/a><p id=\"caption-attachment-9330\" class=\"wp-caption-text\">As tr\u00eas ra\u00edzes c\u00fabicas de $z = 1$.<\/p><\/div>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9325' onClick='GTTabs_show(0,9325)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado ${w_1} = \\frac{{ &#8211; 1 + \\sqrt 3 i}}{2}$ \u00e9 uma raiz c\u00fabica de um n\u00famero complexo $z$. Determine as outras ra\u00edzes c\u00fabicas de $z$. Determine $z$. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21080,"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-9325","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":5581,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12V3Pag143-58_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9325","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=9325"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9325\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21080"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9325"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}