{"id":11093,"date":"2012-10-30T01:50:06","date_gmt":"2012-10-30T01:50:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11093"},"modified":"2022-01-02T12:51:11","modified_gmt":"2022-01-02T12:51:11","slug":"regularidades-com-potencias","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11093","title":{"rendered":"Regularidades com pot\u00eancias"},"content":{"rendered":"<p><ul id='GTTabs_ul_11093' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11093' class='GTTabs_curr'><a  id=\"11093_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11093' ><a  id=\"11093_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_11093'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Regularidades com pot\u00eancias.<\/p>\n<ol>\n<li>Indica o algarismo das unidades de ${11^{153}}$ e de ${2^{22}}$.<\/li>\n<li>Quais s\u00e3o os dois \u00faltimos algarismos da pot\u00eancia ${6^{94}}$? Justifica a tua resposta.<\/li>\n<li>Qual a menor pot\u00eancia de base 2 que termina em 2?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11093' onClick='GTTabs_show(1,11093)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11093'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Comecemos por calcular as primeiras pot\u00eancias de base $11$:<br \/>\n$$\\begin{array}{*{20}{c}} \u00a0 {Expoente:}&amp;1&amp;{}&amp;2&amp;{}&amp;3&amp;{}&amp;4&amp;{}&amp;5 \\\\ \u00a0 {Pot\u00eancia:}&amp;{{{11}^1} = 11}&amp;{}&amp;{{{11}^2} = 121}&amp;{}&amp;{{{11}^3} = 1331}&amp;{}&amp;{{{11}^4} = 14641}&amp;{}&amp;{{{11}^5} = 161051} \\end{array}$$<br \/>\nNo exemplo apresentado, reparamos que o algarismo das unidades \u00e9 $1$. Ser\u00e1 coincid\u00eancia?<\/p>\n<p>N\u00e3o, n\u00e3o \u00e9 coincid\u00eancia. A pot\u00eancia ${{{11}^1} = 11}$. Para calcularmos ${{{11}^2}}$, temos de multiplicar ${{{11}^1} = 11}$ por $11$. Isto \u00e9, ${11^2} = 11 \\times 11$. Ora, o algarismo das unidades do multiplicando \u00e9 $1$ e o algarismo das unidades do multiplicador\u00a0\u00e9 tamb\u00e9m $1$. Logo, o algarismo das unidades do produto \u00e9 $1$, pois $1 \\times 1 = 1$.<\/p>\n<p>Ora, esta rela\u00e7\u00e3o propaga-se de igual forma no c\u00e1lculo das sucessivas pot\u00eancias da base $11$.<\/p>\n<p>Logo, o algarismo das unidades de ${11^{153}}$ \u00e9 $1$.<\/p>\n<p><strong>Curiosidade<\/strong>:<br \/>\nApresenta-se de seguida\u00a0 o valor da pot\u00eancia ${11^{153}}$:<\/p>\n<p><span style=\"color: #0000ff;\">2153182439418108835992644412729930590457960981217217920404787646028283627969774739067090294471817560382112235760043481754894971704029560339644377201975497940331<\/span><\/p>\n<p>De modo an\u00e1logo, temos para as pot\u00eancias de base $2$:<br \/>\n$$\\begin{array}{*{20}{c}} \u00a0 {Expoente:}&amp;1&amp;{}&amp;2&amp;{}&amp;3&amp;{}&amp;4&amp;{}&amp;5&amp;{}&amp;6&amp;{}&amp;7&amp;{}&amp;8&amp;{}&amp;9&amp;{}&amp;{10} \\\\ \u00a0 {Pot\u00eancia:}&amp;{{2^1} = 2}&amp;{}&amp;{{2^2} = 4}&amp;{}&amp;{{2^3} = 8}&amp;{}&amp;{{2^4} = 16}&amp;{}&amp;{{2^5} = 32}&amp;{}&amp;{{2^6} = 64}&amp;{}&amp;{{2^7} = 128}&amp;{}&amp;{{2^8} = 256}&amp;{}&amp;{{2^9} = 512}&amp;{}&amp;{{2^{10}} = 1024} \\end{array}$$<\/p>\n<p>Ora, quando o expoente \u00e9 m\u00faltiplo de $4$ (4, 8, 12, 16, 20, 24, &#8230;), a pot\u00eancia apresenta como algarismo das unidades o algarismo $6$.<\/p>\n<p>Logo, a pot\u00eancia ${2^{20}}$ apresenta $6$ como algarismo das unidades, a pot\u00eancia ${2^{21}}$ apresenta o algarismo $2$ como algarismo das unidades e a pot\u00eancia ${2^{22}}$ apresenta o algarismo $4$ como algarismo das unidades.<\/p>\n<\/li>\n<li>Registando as primeiras pot\u00eancias de base $6$, temos:<br \/>\n$$\\begin{array}{*{20}{c}} \u00a0 {Expoente:}&amp;1&amp;{}&amp;2&amp;{}&amp;3&amp;{}&amp;4&amp;{}&amp;5 \\\\ \u00a0 {Pot\u00eancia:}&amp;{{6^1} = 6}&amp;{}&amp;{{6^2} = 36}&amp;{}&amp;{{6^3} = 216}&amp;{}&amp;{{6^4} = 1296}&amp;{}&amp;{{6^5} = 7776} \\\\ \u00a0 {}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{} \\\\ \u00a0 {Expoente:}&amp;6&amp;{}&amp;7&amp;{}&amp;8&amp;{}&amp;9&amp;{}&amp;{10} \\\\ \u00a0 {Pot\u00eancia:}&amp;{{6^6} = 46656}&amp;{}&amp;{{6^7} = 279936}&amp;{}&amp;{{6^8} = 1679616}&amp;{}&amp;{{6^9} = 10077696}&amp;{}&amp;{{6^{10}} = 60466176} \\\\ \u00a0 {}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{} \\\\ \u00a0 {Expoente:}&amp;{&#8230;}&amp;{}&amp;{&#8230;}&amp;{}&amp;{&#8230;}&amp;{}&amp;{94}&amp;{}&amp;{&#8230;} \\\\ \u00a0 {Pot\u00eancia:}&amp;{&#8230;}&amp;{}&amp;{&#8230;}&amp;{}&amp;{&#8230;}&amp;{}&amp;{{6^{94}} = &#8230;&#8230;96}&amp;{}&amp;{&#8230;} \\end{array}$$<\/p>\n<p>A resposta \u00e9 \u00f3bvia: \u00a0os dois \u00faltimos algarismos da pot\u00eancia ${6^{94}}$ s\u00e3o $96$.<\/p>\n<p><strong>Curiosidade<\/strong>:<br \/>\nApresenta-se de seguida\u00a0 o valor da pot\u00eancia ${6^{94}}$:<\/p>\n<p><span style=\"color: #0000ff;\">14002885448818392191715755040253296907946324384279725470316185836108906496<\/span><\/p>\n<\/p>\n<\/li>\n<li>\n<p>A menor pot\u00eancia de base 2 que termina em 2 \u00e9 ${2^1}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11093' onClick='GTTabs_show(0,11093)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Regularidades com pot\u00eancias. Indica o algarismo das unidades de ${11^{153}}$ e de ${2^{22}}$. Quais s\u00e3o os dois \u00faltimos algarismos da pot\u00eancia ${6^{94}}$? Justifica a tua resposta. Qual a menor pot\u00eancia de&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19267,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[317,97,318],"tags":[428,142],"series":[],"class_list":["post-11093","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-7-o-ano","category-aplicando","category-numeros-inteiros","tag-7-o-ano","tag-potencias"],"views":3790,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat88.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11093","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=11093"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11093\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19267"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11093"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11093"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11093"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11093"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}