{"id":226279,"date":"2025-10-25T15:41:34","date_gmt":"2025-10-25T18:41:34","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=226279"},"modified":"2025-10-25T15:41:34","modified_gmt":"2025-10-25T18:41:34","slug":"problema-elevando-a-9","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-elevando-a-9\/","title":{"rendered":".Problema: Elevando a 9"},"content":{"rendered":"

\n

\n

Problema
\n (Indicado a partir do 1\u00ba ano do E. M.)<\/span><\/p>\n

\n

Dado [tex]x^2+\\dfrac{1}{x^2}=4[\/tex], com [tex]x[\/tex] um n\u00famero real positivo, determine o valor de [tex]x^9+\\dfrac{1}{x^9}.[\/tex]<\/p>\n

Adaptado de @MathCeyhun<\/strong>.<\/p>\n<\/div>\n

\n

\n

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

\n

Inicialmente, observe que, como [tex]x[\/tex] um n\u00famero real positivo, <\/p>\n

[tex]\\qquad x^2+\\dfrac{1}{x^2}=4\\Rightarrow \\left(x+\\dfrac{1}{x}\\right)^2-2=4 \\Rightarrow x+\\dfrac{1}{x}=\\sqrt{6}.[\/tex]<\/p>\n

Por outro lado,<\/p>\n

[tex]\\qquad \\left(x+\\dfrac{1}{x}\\right)^3=x^3+3\\left(x+\\dfrac{1}{x}\\right)+\\dfrac{1}{x^3}\\Rightarrow x^3+\\dfrac{1}{x^3}=\\left(x+\\dfrac{1}{x}\\right)^3-3\\left(x+\\dfrac{1}{x}\\right).[\/tex]<\/p>\n

Dessa forma, <\/p>\n

[tex]\\qquad x^3+\\dfrac{1}{x^3}=(\\sqrt{6})^3-3\\cdot (\\sqrt{6})=3\\cdot \\sqrt{6}.[\/tex]<\/p>\n

Finalmente, <\/p>\n

[tex]\\qquad \\left(x^3+\\dfrac{1}{x^3}\\right)^3=x^9+3\\left(x^3+\\dfrac{1}{x^3}\\right)+\\dfrac{1}{x^9}\\Rightarrow[\/tex]<\/p>\n

[tex]\\qquad x^9+\\dfrac{1}{x^9}=\\left(x^3+\\dfrac{1}{x^3}\\right)^3-3\\left(x^3+\\dfrac{1}{x^3}\\right) \\Rightarrow x^9+\\dfrac{1}{x^9}=(3\\sqrt{6})^3-3\\cdot (3\\cdot \\sqrt{6})\\Rightarrow[\/tex]<\/p>\n

[tex]\\qquad x^9+\\dfrac{1}{x^9}=153\\sqrt{6}.[\/tex]<\/p>\n

\n

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

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 1\u00ba ano do E. M.) Dado [tex]x^2+\\dfrac{1}{x^2}=4[\/tex], com [tex]x[\/tex] um n\u00famero real positivo, determine o valor de [tex]x^9+\\dfrac{1}{x^9}.[\/tex] Adaptado de @MathCeyhun. Solu\u00e7\u00e3o Inicialmente, observe que, como [tex]x[\/tex] um n\u00famero real positivo, [tex]\\qquad x^2+\\dfrac{1}{x^2}=4\\Rightarrow \\left(x+\\dfrac{1}{x}\\right)^2-2=4 \\Rightarrow x+\\dfrac{1}{x}=\\sqrt{6}.[\/tex] Por outro lado, [tex]\\qquad \\left(x+\\dfrac{1}{x}\\right)^3=x^3+3\\left(x+\\dfrac{1}{x}\\right)+\\dfrac{1}{x^3}\\Rightarrow x^3+\\dfrac{1}{x^3}=\\left(x+\\dfrac{1}{x}\\right)^3-3\\left(x+\\dfrac{1}{x}\\right).[\/tex] Dessa forma, [tex]\\qquad x^3+\\dfrac{1}{x^3}=(\\sqrt{6})^3-3\\cdot (\\sqrt{6})=3\\cdot \\sqrt{6}.[\/tex] Finalmente, [tex]\\qquad … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-226279","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/226279","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=226279"}],"version-history":[{"count":7,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/226279\/revisions"}],"predecessor-version":[{"id":226901,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/226279\/revisions\/226901"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=226279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}