{"id":9338,"date":"2012-05-23T00:08:14","date_gmt":"2012-05-22T23:08:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9338"},"modified":"2022-01-27T00:37:22","modified_gmt":"2022-01-27T00:37:22","slug":"as-raizes-quartas-de-1","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9338","title":{"rendered":"As ra\u00edzes quartas de $1$"},"content":{"rendered":"<p><ul id='GTTabs_ul_9338' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9338' class='GTTabs_curr'><a  id=\"9338_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9338' ><a  id=\"9338_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_9338'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Determine as ra\u00edzes quartas de $1$ e represente os seus afixos do diagrama de Argand.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9338' onClick='GTTabs_show(1,9338)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9338'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Como $z = 1 = \\operatorname{cis} \\left( 0 \\right)$, ent\u00e3o as suas ra\u00edzes quartas s\u00e3o:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{k = 0:}&amp;{{w_0} = \\operatorname{cis} \\left( {\\frac{0}{4}} \\right) = \\operatorname{cis} \\left( 0 \\right) = 1} \\\\<br \/>\n{k = 1:}&amp;{{w_1} = \\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{2\\pi }}{4}} \\right) = \\operatorname{cis} \\left( {\\frac{\\pi }{2}} \\right) = i} \\\\<br \/>\n{k = 2:}&amp;{{w_2} = \\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{4\\pi }}{4}} \\right) = \\operatorname{cis} \\left( \\pi\u00a0 \\right) =\u00a0 &#8211; 1} \\\\<br \/>\n{k = 3:}&amp;{{w_3} = \\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{6\\pi }}{4}} \\right) = \\operatorname{cis} \\left( {\\frac{{3\\pi }}{2}} \\right) =\u00a0 &#8211; i}<br \/>\n\\end{array}$$<\/p>\n<\/p>\n<div id=\"attachment_9343\" style=\"width: 467px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9343\" data-attachment-id=\"9343\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9343\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60.png\" data-orig-size=\"761,728\" 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 quartas de $z = 1$.&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60.png\" class=\" wp-image-9343 \" title=\"Plano de Argand\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60.png\" alt=\"\" width=\"457\" height=\"437\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60.png 761w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60-300x286.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60-150x143.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12pag143-60-400x382.png 400w\" sizes=\"auto, (max-width: 457px) 100vw, 457px\" \/><\/a><p id=\"caption-attachment-9343\" class=\"wp-caption-text\">As ra\u00edzes quartas de $z = 1$.<\/p><\/div>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9338' onClick='GTTabs_show(0,9338)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine as ra\u00edzes quartas de $1$ e represente os seus afixos do diagrama de Argand. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":21081,"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-9338","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":1594,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/05\/12V3Pag143-60_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9338","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=9338"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9338\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21081"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9338"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9338"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9338"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}