{"id":39716,"date":"2015-06-10T21:52:49","date_gmt":"2015-06-11T00:52:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=39716"},"modified":"2025-06-24T01:43:24","modified_gmt":"2025-06-24T04:43:24","slug":"problema-e-radical-simplificar","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-e-radical-simplificar\/","title":{"rendered":".Problema: \u00c9 radical simplificar"},"content":{"rendered":"

\n

\n

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

\nSimplifique a express\u00e3o:<\/p>\n

[tex]\\;\\sqrt{2+\\sqrt{3}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{3}}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}} \\cdot \\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}[\/tex].\n\t<\/p><\/div>\n

\n\"explicador_p\" <\/p>\n

Lembretes <\/p>\n

\n

✐ <\/span>[tex]\\sqrt{a} \\cdot\\sqrt{b}=\\sqrt{a\\cdot b} [\/tex], para quaisquer n\u00fameros reais n\u00e3o negativos [tex]a\\,[\/tex] e [tex]\\,b.[\/tex]<\/p>\n

✐ <\/span> Produto da soma pela diferen\u00e7a:
\n[tex]\\qquad \\boxed{(m+n) \\cdot (m-n)=m^2-n^2}[\/tex], para quaisquer [tex]m,n\\in\\mathbb{R}\\,.[\/tex]<\/div>\n

\n

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

\nMultiplicamos, inicialmente, os dois \u00faltimos radicais:<\/p>\n

[tex]\\qquad \\begin{align*}
\n&\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}} \\cdot \\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}=\\\\
\n&=\\sqrt{\\left(2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}\\right) \\cdot \\left(2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}\\right)}\\\\
\n&=\\sqrt{2^2-\\left(\\sqrt{2+\\sqrt{2+\\sqrt{3}}}\\right)^2~} \\\\
\n&=\\sqrt{4-\\left(2+\\sqrt{2+\\sqrt{3}}\\right)}\\\\
\n&=\\sqrt{2-\\sqrt{2+\\sqrt{3}}}\\,.\\end{align*}[\/tex]<\/p>\n

Agora, multipliquemos o segundo fator da express\u00e3o inicial por esse resultado encontrado:<\/p>\n

[tex]\\qquad \\begin{align*}
\n\\sqrt{2+\\sqrt{2+\\sqrt{3}}} \\cdot \\sqrt{2-\\sqrt{2+\\sqrt{3}}}&= \\sqrt{\\left(2+\\sqrt{2+\\sqrt{3}}\\right) \\cdot \\left(2-\\sqrt{2+\\sqrt{3}}\\right)}\\\\
\n&= \\sqrt{2^2-\\left(\\sqrt{2+\\sqrt{3}}\\right)^2}\\\\
\n&= \\sqrt{4-\\left(2+\\sqrt{3}\\right)}\\\\
\n&=\\sqrt{2-\\sqrt{3}}\\,.\\end{align*}[\/tex]<\/p>\n

Finalmente, multipliquemos o primeiro fator da express\u00e3o dada por esse segundo resultado obtido:<\/p>\n

[tex]\\qquad \\sqrt{2+\\sqrt{3}} \\cdot \\sqrt{2-\\sqrt{3}}=1[\/tex].<\/p>\n

\tPortanto:
\n\t[tex]\\;\\sqrt{2+\\sqrt{3}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{3}}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}} \\cdot \\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}=1[\/tex].<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) Simplifique a express\u00e3o: [tex]\\;\\sqrt{2+\\sqrt{3}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{3}}} \\cdot \\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}} \\cdot \\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}[\/tex]. Lembretes ✐ [tex]\\sqrt{a} \\cdot\\sqrt{b}=\\sqrt{a\\cdot b} [\/tex], para quaisquer n\u00fameros reais n\u00e3o negativos [tex]a\\,[\/tex] e [tex]\\,b.[\/tex] ✐ Produto da soma pela diferen\u00e7a: [tex]\\qquad \\boxed{(m+n) \\cdot (m-n)=m^2-n^2}[\/tex], para quaisquer [tex]m,n\\in\\mathbb{R}\\,.[\/tex] Solu\u00e7\u00e3o Multiplicamos, inicialmente, os dois \u00faltimos radicais: … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-39716","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/39716","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=39716"}],"version-history":[{"count":33,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/39716\/revisions"}],"predecessor-version":[{"id":226035,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/39716\/revisions\/226035"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=39716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}