Ensino Médio ⇒ Radical Duplo Tópico resolvido

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Verifique a identidade
[tex3]\sqrt{2}.\sqrt{2+\sqrt{2}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}=2[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\sqrt{2}.\sqrt{2+\sqrt{2}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}=2[/tex3]

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Obrigada! Desse jeito realmente é mais fácil.

[csmarcelo]

Eu estava no celular e acabei excluindo a mensagem, pois vi que ficou incompleta. Segue o desenvolvimento novamente, agora completo.

[tex3]\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=2[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=2[/tex3]

[tex3]{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=\sqrt{\(2+\sqrt{2+\sqrt{2}}\)\(2-\sqrt{2+\sqrt{2}}\)}=\sqrt{2^2-\(\sqrt{2+\sqrt{2}}\)^2}=\sqrt{2-(2+\sqrt{2})}={\color{blue}\sqrt{2-\sqrt{2}}}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=\sqrt{\(2+\sqrt{2+\sqrt{2}}\)\(2-\sqrt{2+\sqrt{2}}\)}=\sqrt{2^2-\(\sqrt{2+\sqrt{2}}\)^2}=\sqrt{2-(2+\sqrt{2})}={\color{blue}\sqrt{2-\sqrt{2}}}[/tex3]

Logo,

[tex3]\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2-\sqrt{2}}}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}}=\sqrt{2}\cdot\sqrt{2+\sqrt{2}}\cdot{\color{blue}\sqrt{2-\sqrt{2}}}[/tex3]

[tex3]\sqrt{2}\cdot{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=2[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\sqrt{2}\cdot{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=2[/tex3]

[tex3]{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=\sqrt{(2+\sqrt{2})(2-\sqrt{2})}=\sqrt{2^2-\sqrt{2}^2}={\color{red}\sqrt{2}}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=\sqrt{(2+\sqrt{2})(2-\sqrt{2})}=\sqrt{2^2-\sqrt{2}^2}={\color{red}\sqrt{2}}[/tex3]

Logo,

[tex3]\sqrt{2}\cdot{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=\sqrt{2}\cdot{\color{red}\sqrt{2}}=2[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\sqrt{2}\cdot{\color{red}\sqrt{2+\sqrt{2}}\cdot\sqrt{2-\sqrt{2}}}=\sqrt{2}\cdot{\color{red}\sqrt{2}}=2[/tex3]