{"id":107,"date":"2020-03-23T16:22:27","date_gmt":"2020-03-23T19:22:27","guid":{"rendered":"http:\/\/cinoto.com.br\/matematica\/?p=107"},"modified":"2020-03-23T16:26:36","modified_gmt":"2020-03-23T19:26:36","slug":"se-5%c2%b2-9%c2%b2-12%c2%b2-17%c2%b2-for-escrito-sob-a-forma-a%c2%b2-b%c2%b2","status":"publish","type":"post","link":"http:\/\/cinoto.com.br\/matematica\/se-5%c2%b2-9%c2%b2-12%c2%b2-17%c2%b2-for-escrito-sob-a-forma-a%c2%b2-b%c2%b2\/","title":{"rendered":"3) Se (5\u00b2 + 9\u00b2).(12\u00b2 + 17\u00b2) for escrito sob a forma a\u00b2 + b\u00b2, com a e b n\u00fameros naturais, ent\u00e3o quanto vale a + b?"},"content":{"rendered":"\n\n\n\n\n

Resolu\u00e7\u00e3o:<\/h2>\n\n\n\n

Essa foi dif\u00edcil. Agrade\u00e7o a ajuda dos colegas da lista de discuss\u00e3o da Olimp\u00edada Brasileira de Matem\u00e1tica.<\/p>\n\n\n\n

Trabalhando com n\u00fameros complexos, sabemos que o m\u00f3dulo do produto de dois complexos ao quadrado \u00e9 igual ao produto do quadrado de cada um. Seja z e w dois complexos, ent\u00e3o temos:<\/p>\n\n\n\n

|zw|\u00b2 = |z|\u00b2 . |w|\u00b2<\/p>\n\n\n\n

Ent\u00e3o considere os complexos:<\/p>\n\n\n\n

z = 5 + 9i<\/p>\n\n\n\n

w = 12 + 17i<\/p>\n\n\n\n

E voc\u00ea ter\u00e1:<\/p>\n\n\n\n

|zw|\u00b2 = |z|\u00b2 . |w|\u00b2<\/p>\n\n\n\n

|(5 + 9i).(12 + 17i)|\u00b2 = |5 + 9i|\u00b2 . |12 + 17i|\u00b2<\/p>\n\n\n\n

|60 + 85i + 108i + 153i\u00b2|\u00b2 = [raiz(5\u00b2 + 9\u00b2)]\u00b2 . [raiz(12\u00b2 + 17\u00b2)]\u00b2<\/p>\n\n\n\n

|60 + 193i – 153|\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

|-93 + 193i|\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

[raiz(93\u00b2 + 193\u00b2)]\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

93\u00b2 + 193\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

Ent\u00e3o a + b = 93 + 193 = 286<\/p>\n\n\n\n

Mas s\u00f3 que poder\u00edamos escrever tamb\u00e9m de outra forma:<\/p>\n\n\n\n

z = 9 + 5i<\/p>\n\n\n\n

w = 12 + 17i<\/p>\n\n\n\n

E voc\u00ea ter\u00e1:<\/p>\n\n\n\n

|zw|\u00b2 = |z|\u00b2 . |w|\u00b2<\/p>\n\n\n\n

|(9 + 5i).(12 + 17i)|\u00b2 = |9 + 5i|\u00b2 . |12 + 17i|\u00b2<\/p>\n\n\n\n

|108 + 153i + 60i + 85i\u00b2|\u00b2 = [raiz(9\u00b2 + 5\u00b2)]\u00b2 . [raiz(12\u00b2 + 17\u00b2)]\u00b2<\/p>\n\n\n\n

|108 + 213i – 85|\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

|23 + 213i|\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

[raiz(23\u00b2 + 213\u00b2)]\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

23\u00b2 + 213\u00b2 = (5\u00b2 + 9\u00b2) . (12\u00b2 + 17\u00b2)<\/p>\n\n\n\n

Ent\u00e3o a + b = 23 + 213 = 236<\/p>\n\n\n\n

Voc\u00ea poderia escrever w de outra forma tamb\u00e9m, mas a\u00ed cair\u00edamos nas mesmas solu\u00e7\u00f5es.<\/p>\n\n\n\n

Resposta: a + b = 286 ou a + b = 236.<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[30],"class_list":["post-107","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-insano"],"_links":{"self":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/107","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/comments?post=107"}],"version-history":[{"count":1,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/107\/revisions"}],"predecessor-version":[{"id":108,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/107\/revisions\/108"}],"wp:attachment":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/media?parent=107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/categories?post=107"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/tags?post=107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}