\nEnunciado<\/b><\/span><\/p>\n
A partir de ${i^2} =\u00a0 – 1$<\/p>\n
\n
Calcule: ${i^3}$, ${i^4}$, ${i^6}$, ${i^{10}}$, ${i^{96}}$ e ${i^{105}}$.<\/li>\n
Para todo o $n \\in \\mathbb{N}$, calcule: ${i^{4n}}$, ${i^{4n + 1}}$, ${i^{4n + 2}}$ e ${i^{4n + 3}}$.<\/li>\n<\/ol>\n
Resolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n
\nResolu\u00e7\u00e3o<\/b><\/span><\/p>\n
\n
Ora,
\n$$\\begin{array}{*{20}{l}}
\n{{i^3} = {i^2} \\times i =\u00a0 – 1 \\times i =\u00a0 – i} \\\\
\n{{i^4} = {{\\left( {{i^2}} \\right)}^2} = {{( – 1)}^2} = 1} \\\\
\n{{i^6} = {{\\left( {{i^2}} \\right)}^3} = {{( – 1)}^3} =\u00a0 – 1} \\\\
\n{{i^7} = {{\\left( {{i^2}} \\right)}^3} \\times i = {{( – 1)}^3} \\times i =\u00a0 – i} \\\\
\n{{i^{10}} = {{\\left( {{i^2}} \\right)}^4} \\times {i^2} = {{( – 1)}^4} \\times ( – 1) =\u00a0 – 1} \\\\
\n{{i^{96}} = {{\\left( {{i^2}} \\right)}^{48}} = {{( – 1)}^{48}} = 1} \\\\
\n{{i^{105}} = {{\\left( {{i^2}} \\right)}^{52}} \\times i = {{( – 1)}^{52}} \\times i = i}
\n\\end{array}$$<\/li>\n
Ora,
\n$$\\begin{array}{*{20}{l}}
\n{{i^{4n}} = {{\\left( {{i^4}} \\right)}^n} = {1^n} = 1,\\forall n \\in \\mathbb{N}} \\\\
\n{{i^{4n + 1}} = {{\\left( {{i^4}} \\right)}^n} \\times i = {1^n} \\times i = i,\\forall n \\in \\mathbb{N}} \\\\
\n{{i^{4n + 2}} = {{\\left( {{i^4}} \\right)}^n} \\times {i^2} = {1^n} \\times ( – 1) =\u00a0 – 1,\\forall n \\in \\mathbb{N}} \\\\
\n{{i^{4n + 3}} = {{\\left( {{i^4}} \\right)}^n} \\times {i^3} = {1^n} \\times ( – i) =\u00a0 – i,\\forall n \\in \\mathbb{N}}
\n\\end{array}$$<\/li>\n<\/ol>\n
<< Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"
Enunciado Resolu\u00e7\u00e3o Enunciado A partir de ${i^2} =\u00a0 – 1$ Calcule: ${i^3}$, ${i^4}$, ${i^6}$, ${i^{10}}$, ${i^{96}}$ e ${i^{105}}$. Para todo o $n \\in \\mathbb{N}$, calcule: ${i^{4n}}$, ${i^{4n + 1}}$, ${i^{4n + 2}}$ e ${i^{4n...<\/p>\n","protected":false},"author":1,"featured_media":19554,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18,311],"series":[],"class_list":["post-8912","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","tag-unidade-imaginaria"],"views":1746,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat173.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8912","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=8912"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8912\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19554"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8912"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8912"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8912"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}