{"id":9290,"date":"2012-05-22T14:48:58","date_gmt":"2012-05-22T13:48:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9290"},"modified":"2021-12-29T22:25:11","modified_gmt":"2021-12-29T22:25:11","slug":"prove-que-w-1-i-e-uma-raiz-cubica-de-z-2-2i","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9290","title":{"rendered":"Prove que $w = 1 + i$ \u00e9 uma raiz c\u00fabica de $z =  &#8211; 2 + 2i$"},"content":{"rendered":"<p><ul id='GTTabs_ul_9290' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9290' class='GTTabs_curr'><a  id=\"9290_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9290' ><a  id=\"9290_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_9290'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Prove que $w = 1 + i$ \u00e9 uma raiz c\u00fabica de $z =\u00a0 &#8211; 2 + 2i$.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9290' onClick='GTTabs_show(1,9290)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9290'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{w^3}}&amp; = &amp;{{{\\left( {1 + i} \\right)}^3}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {\\sqrt 2 \\operatorname{cis} \\frac{\\pi }{4}} \\right)}^3}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {\\sqrt 2 } \\right)}^3}\\operatorname{cis} \\left( {\\frac{{3\\pi }}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2\\sqrt 2 \\left( {\\cos \\left( {\\frac{{3\\pi }}{4}} \\right) + i\\operatorname{sen} \\left( {\\frac{{3\\pi }}{4}} \\right)} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2\\sqrt 2 \\left( { &#8211; \\frac{{\\sqrt 2 }}{2} + \\frac{{\\sqrt 2 }}{2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 2 + 2i} \\\\<br \/>\n{}&amp; = &amp;z<br \/>\n\\end{array}$$<\/p>\n<p>Como ${w^3} = z$, ent\u00e3o $w$ \u00e9 uma raiz c\u00fabica de $z$.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9290' onClick='GTTabs_show(0,9290)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Prove que $w = 1 + i$ \u00e9 uma raiz c\u00fabica de $z =\u00a0 &#8211; 2 + 2i$. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19171,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,314,18],"series":[],"class_list":["post-9290","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-forma-trigonometrica","tag-numeros-complexos"],"views":1688,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat62.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9290","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=9290"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9290\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19171"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9290"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}