{"id":187707,"date":"2023-05-02T22:40:26","date_gmt":"2023-05-03T01:40:26","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=187707"},"modified":"2023-05-21T00:38:22","modified_gmt":"2023-05-21T03:38:22","slug":"probleminha-potencia-complicada-s","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/probleminha-potencia-complicada-s\/","title":{"rendered":".Probleminha: Pot\u00eancia complicada"},"content":{"rendered":"

✏ Link do problema para dispositivos da Apple<\/em>.<\/span><\/a><\/strong><\/p>\n

\n

Problema
\n (Indicado a partir do 8\u00ba ano do E. F.)<\/span><\/p>\n

\n(Portal da Matem\u00e1tica<\/strong> – Adaptado) <\/span> Sejam [tex]\\boxed{a = \\left(\\frac{1}{10}\\right)^{-2}-\\left(\\frac{1}{6}\\right)^{-2}}\\,[\/tex] e [tex]\\, \\boxed{b = \\dfrac{2\\cdot \\left(\\frac{1}{3}\\right)^{-1}-2^2}{\\left(\\frac{1}{2}\\right)^{-2}}}[\/tex].
\nDetermine o valor de [tex]a^b[\/tex].<\/strong>\n<\/div>\n

\n

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

Lembretes <\/p>\n

\n

(I)<\/strong> <\/span> Se [tex]a[\/tex] e [tex]x[\/tex] s\u00e3o n\u00fameros reais positivos, ent\u00e3o [tex]a^{-x} = \\dfrac{1}{a^x}[\/tex].
\n (II)<\/strong> <\/span> Se [tex]a[\/tex] e [tex]b[\/tex] s\u00e3o n\u00fameros reais positivos, ent\u00e3o [tex]\\left(\\dfrac{a}{b}\\right)^x = \\dfrac{a^x}{b^x}[\/tex].\n<\/div>\n

\n

\n

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

\nInicialmente vamos encontrar o valor num\u00e9rico de [tex]a[\/tex]:<\/p>\n

[tex]\\qquad a = \\left(\\frac{1}{10}\\right)^{-2}-\\left(\\frac{1}{6}\\right)^{-2}[\/tex]<\/p>\n

[tex]\\qquad a = \\dfrac{1}{\\left(\\frac{1}{10}\\right)^{2}}-\\dfrac{1}{\\left(\\frac{1}{6}\\right)^{2}}[\/tex]<\/p>\n

[tex]\\qquad a = \\dfrac{1}{\\frac{1}{100}}-\\dfrac{1}{\\frac{1}{36}}[\/tex]<\/p>\n

[tex]\\qquad a = 100-36[\/tex]<\/p>\n

[tex]\\qquad \\boxed{a = 64}\\;.[\/tex]<\/p>\n

O valor num\u00e9rico de [tex]b[\/tex] \u00e9 dado por:<\/p>\n

[tex]\\qquad b = \\dfrac{2\\cdot \\dfrac{1}{\\frac{1}{3}}-2^2}{\\dfrac{1}{\\left(\\frac{1}{2}\\right)^{2}}}[\/tex]<\/p>\n

[tex]\\qquad b = \\dfrac{2\\cdot 3-2^2}{\\dfrac{1}{\\frac{1}{4}}}[\/tex]<\/p>\n

[tex]\\qquad b = \\dfrac{2\\cdot 3-2^2}{4}[\/tex]<\/p>\n

[tex]\\qquad \\boxed{b = \\dfrac{1}{2}}\\;.[\/tex]<\/p>\n

Logo,<\/p>\n

[tex]\\qquad a^b = 64^{\\frac{1}{2}}[\/tex]<\/p>\n

[tex]\\qquad a^b = \\sqrt{64}[\/tex]<\/p>\n

[tex]\\qquad \\fcolorbox{black}{#eee0e5}{$a^b = 8$}\\,.[\/tex]<\/p>\n

\n

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

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

✏ Link do problema para dispositivos da Apple. Problema (Indicado a partir do 8\u00ba ano do E. F.) (Portal da Matem\u00e1tica – Adaptado) Sejam [tex]\\boxed{a = \\left(\\frac{1}{10}\\right)^{-2}-\\left(\\frac{1}{6}\\right)^{-2}}\\,[\/tex] e [tex]\\, \\boxed{b = \\dfrac{2\\cdot \\left(\\frac{1}{3}\\right)^{-1}-2^2}{\\left(\\frac{1}{2}\\right)^{-2}}}[\/tex]. Determine o valor de [tex]a^b[\/tex]. Lembretes (I) Se [tex]a[\/tex] e [tex]x[\/tex] s\u00e3o n\u00fameros reais positivos, ent\u00e3o [tex]a^{-x} = \\dfrac{1}{a^x}[\/tex]. (II) Se [tex]a[\/tex] … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":254,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-187707","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187707","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\/254"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=187707"}],"version-history":[{"count":8,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187707\/revisions"}],"predecessor-version":[{"id":191201,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187707\/revisions\/191201"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=187707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}