{"id":127908,"date":"2019-05-15T20:52:27","date_gmt":"2019-05-15T23:52:27","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=127908"},"modified":"2023-05-31T04:39:39","modified_gmt":"2023-05-31T07:39:39","slug":"probleminha-quantidade-de-zeros","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/probleminha-quantidade-de-zeros\/","title":{"rendered":".Probleminha: Quantidade de zeros"},"content":{"rendered":"

\n

\n

Problema
\n (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n

\n

Considere o n\u00famero [tex]8^{673} \\cdot 625^{505}[\/tex]. Quantos zeros aparecem em sua representa\u00e7\u00e3o decimal?<\/p>\n

\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o <\/p>\n

\n

Observe que:
\n [tex] \\begin{align*}8^{673} \\cdot 625^{505} &= (2^3)^{673} \\cdot (5^4)^{505}\\\\
\n&= 2^{3\\cdot 673} \\cdot 5^{4\\cdot 505}\\\\
\n&= 2^{2019} \\cdot 5^{2020}\\\\
\n&= 2^{2019} \\cdot 5^{2019} \\cdot 5\\\\
\n&= 5 \\cdot (2 \\cdot 5) ^{2019}\\\\
\n&= 5 \\cdot 10^{2019} \\, \\, .\\end{align*} [\/tex]<\/p>\n

Assim, o n\u00famero [tex]8^{673} \\cdot 625^{505}[\/tex] \u00e9 igual a [tex] 5\\underbrace{000\\cdots 00}_{2019 \\text{ zeros}}[\/tex]; tendo, portanto, [tex] \\fcolorbox{black}{#eee0e5}{$2019$} \\, [\/tex] zeros.<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n

\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) Considere o n\u00famero [tex]8^{673} \\cdot 625^{505}[\/tex]. Quantos zeros aparecem em sua representa\u00e7\u00e3o decimal? Solu\u00e7\u00e3o Observe que: [tex] \\begin{align*}8^{673} \\cdot 625^{505} &= (2^3)^{673} \\cdot (5^4)^{505}\\\\ &= 2^{3\\cdot 673} \\cdot 5^{4\\cdot 505}\\\\ &= 2^{2019} \\cdot 5^{2020}\\\\ &= 2^{2019} \\cdot 5^{2019} \\cdot 5\\\\ &= 5 \\cdot (2 \\cdot … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":736,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-127908","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/127908","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/736"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=127908"}],"version-history":[{"count":7,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/127908\/revisions"}],"predecessor-version":[{"id":192478,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/127908\/revisions\/192478"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=127908"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}