{"id":9351,"date":"2012-05-23T15:27:03","date_gmt":"2012-05-23T14:27:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9351"},"modified":"2022-01-27T00:50:06","modified_gmt":"2022-01-27T00:50:06","slug":"determine-as-raizes-da-equacao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9351","title":{"rendered":"Determine as ra\u00edzes da equa\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_9351' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9351' class='GTTabs_curr'><a  id=\"9351_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9351' ><a  id=\"9351_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_9351'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Dado o n\u00famero complexo $w = 27\\operatorname{cis} \\frac{\\pi }{3}$, determine as ra\u00edzes da equa\u00e7\u00e3o ${z^3} + w = 0$, representando as imagens no plano de Argand.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9351' onClick='GTTabs_show(1,9351)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9351'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{z^3} + 27\\operatorname{cis} \\frac{\\pi }{3} = 0}&amp; \\Leftrightarrow &amp;{{z^3} =\u00a0 &#8211; 27\\operatorname{cis} \\frac{\\pi }{3}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{z^3} = 27\\operatorname{cis} \\left( {\\frac{\\pi }{3} + \\pi } \\right)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = \\sqrt[3]{{27}}\\operatorname{cis} \\left( {\\frac{{\\tfrac{{4\\pi }}{3}}}{3}} \\right)}&amp; \\vee &amp;{z = \\sqrt[3]{{27}}\\operatorname{cis} \\left( {\\frac{{\\tfrac{{4\\pi }}{3}}}{3} + \\frac{{2\\pi }}{3}} \\right)}&amp; \\vee &amp;{z = \\sqrt[3]{{27}}\\operatorname{cis} \\left( {\\frac{{\\tfrac{{4\\pi }}{3}}}{3} + \\frac{{4\\pi }}{3}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 3\\operatorname{cis} \\left( {\\frac{{4\\pi }}{9}} \\right)}&amp; \\vee &amp;{z = 3\\operatorname{cis} \\left( {\\frac{{10\\pi }}{9}} \\right)}&amp; \\vee &amp;{z = 3\\operatorname{cis} \\left( {\\frac{{16\\pi }}{9}} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<div id=\"attachment_9355\" style=\"width: 386px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9355\" data-attachment-id=\"9355\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9355\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62.png\" data-orig-size=\"376,382\" 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=\"Plano de Argand\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;As ra\u00edzes da equa\u00e7\u00e3o ${{z^3} + 27\\operatorname{cis} \\frac{\\pi }{3} = 0}$.&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62.png\" class=\"size-full wp-image-9355\" title=\"Plano de Argand\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62.png\" alt=\"\" width=\"376\" height=\"382\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62.png 376w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62-295x300.png 295w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-62-147x150.png 147w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/a><p id=\"caption-attachment-9355\" class=\"wp-caption-text\">As ra\u00edzes da equa\u00e7\u00e3o \\({z^3} + 27{\\mathop{\\rm cis}\\nolimits} \\frac{\\pi }{3} = 0\\).<\/p><\/div>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9351' onClick='GTTabs_show(0,9351)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Dado o n\u00famero complexo $w = 27\\operatorname{cis} \\frac{\\pi }{3}$, determine as ra\u00edzes da equa\u00e7\u00e3o ${z^3} + w = 0$, representando as imagens no plano de Argand. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":21084,"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-9351","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":3524,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12V3Pag143-62_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9351","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=9351"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9351\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21084"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9351"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9351"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9351"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9351"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}