{"id":9315,"date":"2012-05-22T18:52:24","date_gmt":"2012-05-22T17:52:24","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9315"},"modified":"2021-12-29T22:52:19","modified_gmt":"2021-12-29T22:52:19","slug":"considere-as-equacoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9315","title":{"rendered":"Considere as equa\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_9315' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9315' class='GTTabs_curr'><a  id=\"9315_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9315' ><a  id=\"9315_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_9315'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as equa\u00e7\u00f5es: $$\\begin{array}{*{20}{c}}<br \/>\n{{w^2} = 4}&amp;{\\text{e}}&amp;{{w^4} = 16}<br \/>\n\\end{array}$$<\/p>\n<p>As equa\u00e7\u00f5es dadas s\u00e3o equivalentes em $\\mathbb{R}$? E em $\\mathbb{C}$?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9315' onClick='GTTabs_show(1,9315)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9315'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{w^2} = 4}&amp;{\\text{e}}&amp;{{w^4} = 16}<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<p><strong>Resolvendo em $\\mathbb{R}$<\/strong>:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{w^2} = 4}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w =\u00a0 &#8211; 2}&amp; \\vee &amp;{w = 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{w^4} = 16}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w =\u00a0 &#8211; 2}&amp; \\vee &amp;{w = 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, as equa\u00e7\u00f5es s\u00e3o equivalentes em $\\mathbb{R}$, pois possuem igual conjunto solu\u00e7\u00e3o.<br \/>\n\u00ad<\/p>\n<p><strong>Resolvendo em $\\mathbb{C}$<\/strong>:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{w^2} = 4}&amp; \\Leftrightarrow &amp;{{w^2} = 4\\operatorname{cis} \\left( 0 \\right)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = \\sqrt 4 \\operatorname{cis} \\left( {\\frac{0}{2}} \\right)}&amp; \\vee &amp;{w = \\sqrt 4 \\operatorname{cis} \\left( {\\frac{0}{2} + \\frac{{2\\pi }}{2}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = 2\\operatorname{cis} \\left( 0 \\right)}&amp; \\vee &amp;{w = 2\\operatorname{cis} \\left( \\pi\u00a0 \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = 2}&amp; \\vee &amp;{w =\u00a0 &#8211; 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{w^4} = 16}&amp; \\Leftrightarrow &amp;{{w^4} = 16\\operatorname{cis} \\left( 0 \\right)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = \\sqrt[4]{{16}}\\operatorname{cis} \\left( {\\frac{0}{4}} \\right)}&amp; \\vee &amp;{w = \\sqrt[4]{{16}}\\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{2\\pi }}{4}} \\right)}&amp; \\vee &amp;{w = \\sqrt[4]{{16}}\\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{4\\pi }}{4}} \\right)}&amp; \\vee &amp;{w = \\sqrt[4]{{16}}\\operatorname{cis} \\left( {\\frac{0}{4} + \\frac{{6\\pi }}{4}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = 2\\operatorname{cis} \\left( 0 \\right)}&amp; \\vee &amp;{w = 2\\operatorname{cis} \\left( {\\frac{\\pi }{2}} \\right)}&amp; \\vee &amp;{w = 2\\operatorname{cis} \\left( \\pi\u00a0 \\right)}&amp; \\vee &amp;{w = 2\\operatorname{cis} \\left( {\\frac{{3\\pi }}{2}} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{w = 2}&amp; \\vee &amp;{w = 2i}&amp; \\vee &amp;{w =\u00a0 &#8211; 2}&amp; \\vee &amp;{w =\u00a0 &#8211; 2i}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, as equa\u00e7\u00f5es n\u00e3o s\u00e3o equivalentes em $\\mathbb{C}$, pois possuem diferentes conjuntos solu\u00e7\u00e3o.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9315' onClick='GTTabs_show(0,9315)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere as equa\u00e7\u00f5es: $$\\begin{array}{*{20}{c}} {{w^2} = 4}&amp;{\\text{e}}&amp;{{w^4} = 16} \\end{array}$$ As equa\u00e7\u00f5es dadas s\u00e3o equivalentes em $\\mathbb{R}$? E em $\\mathbb{C}$? Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19170,"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-9315","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":1822,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat61.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9315","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=9315"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9315\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19170"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9315"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}