{"id":223108,"date":"2025-03-12T10:47:33","date_gmt":"2025-03-12T13:47:33","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=223108"},"modified":"2025-03-12T10:47:33","modified_gmt":"2025-03-12T13:47:33","slug":"desafio-raiz-complicada","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/desafio-raiz-complicada\/","title":{"rendered":".Desafio: Raiz complicada"},"content":{"rendered":"
\n

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

\nSem o uso de calculadora, calcule o valor de [tex]\\sqrt{4444488889}[\/tex].<\/p>\n

Extra\u00eddo de Banco de Quest\u00f5es OBMEP<\/strong>.<\/p>\n<\/div>\n

 <\/p>\n

\n

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

\n

Repare que:
\n[tex]\\qquad 4444488889=4444444444+44444+1[\/tex].<\/p>\n

Agora, note que:
\n[tex]\\qquad 4444444444=4\\cdot 1111111111[\/tex].<\/p>\n

Como [tex]\\underbrace{111\\cdots11}_{n~uns}[\/tex] \u00e9 igual a [tex]\\dfrac {10^{n}-1}{9}[\/tex], temos que:<\/p>\n

[tex]\\qquad 4 \\cdot 1111111111= 4 \\cdot \\left(\\dfrac {10^{10}-1}{9}\\right)[\/tex].<\/p>\n

De modo an\u00e1logo:<\/p>\n

[tex]\\qquad 4 \\cdot 11111= 4 \\cdot \\left(\\dfrac {10^{5}-1}{9}\\right)[\/tex].<\/p>\n

Assim, <\/p>\n

[tex]\\qquad 4444488889=4 \\cdot \\left(\\dfrac {10^{10}-1}{9}\\right)+ 4 \\cdot \\left(\\dfrac {10^{5}-1}{9}\\right) +\\dfrac{9}{9}[\/tex]<\/p>\n

[tex]\\qquad =\\dfrac{1}{9} \\cdot (4\\cdot 10^{10} -4+4\\cdot 10^{5} -4+9) [\/tex]<\/p>\n

[tex]\\qquad =\\dfrac{1}{9}\\cdot(4 \\cdot 10^{10}+ 4 \\cdot 10^{5}+1)[\/tex]<\/p>\n

[tex]\\qquad =\\dfrac{1}{9}[(2 \\cdot 10^{5} )^{2}+2 \\cdot 2 \\cdot 10^{5} \\cdot1+1^{2}][\/tex]<\/p>\n

[tex]\\qquad =\\dfrac{1}{9} \\cdot (2 \\cdot10^{5}+1)^{2}[\/tex]<\/p>\n

[tex]\\qquad = \\left(\\dfrac {2\\cdot10^{5}+1}{3}\\right)^{2}[\/tex].<\/p>\n

Portanto,
\n[tex]\\qquad \\sqrt{4444488889}=\\sqrt{\\left(\\dfrac {2\\cdot10^{5}+1}{3}\\right)^{2}} =\\dfrac {2\\cdot10^{5}+1}{3}= \\dfrac {200001}{3}=66667[\/tex].<\/p>\n

\n

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

 <\/p>\n

\nParticipou da discuss\u00e3o o Clube Phidias<\/strong>.<\/div>\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) Sem o uso de calculadora, calcule o valor de [tex]\\sqrt{4444488889}[\/tex]. Extra\u00eddo de Banco de Quest\u00f5es OBMEP.   Solu\u00e7\u00e3o Repare que: [tex]\\qquad 4444488889=4444444444+44444+1[\/tex]. Agora, note que: [tex]\\qquad 4444444444=4\\cdot 1111111111[\/tex]. Como [tex]\\underbrace{111\\cdots11}_{n~uns}[\/tex] \u00e9 igual a [tex]\\dfrac {10^{n}-1}{9}[\/tex], temos que: [tex]\\qquad 4 \\cdot 1111111111= 4 \\cdot \\left(\\dfrac {10^{10}-1}{9}\\right)[\/tex]. … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":668,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-223108","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223108","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\/668"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=223108"}],"version-history":[{"count":10,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223108\/revisions"}],"predecessor-version":[{"id":224816,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/223108\/revisions\/224816"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=223108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}