Resposta:
d)
Explicação passo-a-passo:
Considerando que [tex]a+b=k[/tex]:
[tex]k^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2[/tex]
[tex]k^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}\right)^2+\left(\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2+2\cdot\sqrt{4+\sqrt{10+2\sqrt{5}}}\cdot\sqrt{4-\sqrt{10+2\sqrt{5}}}[/tex]
[tex]k^2=4+\sqrt{10+2\sqrt{5}}}+4-\sqrt{10+2\sqrt{5}}+2\cdot\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\cdot\left(4-\sqrt{10+2\sqrt{5}}\right)}[/tex]
[tex]k^2=8+2\cdot\sqrt{4^2-\left(\sqrt{10+2\sqrt{5}}\right)^2}[/tex]
[tex]k^2=8+2\cdot\sqrt{16-(10+2\sqrt{5})}[/tex]
[tex]k^2=8+2\cdot\sqrt{6-2\sqrt{5}}[/tex]
[tex]k^2-8=2\cdot\sqrt{6-2\sqrt{5}}[/tex]
[tex](k^2-8)^2=\left(2\cdot\sqrt{6-2\sqrt{5}}\right)^2[/tex]
[tex](k^2-8)^2=4\cdot(6-2\sqrt{5})[/tex]
[tex](k^2-8)^2=24-8\sqrt{5}[/tex]
Vamos agora testar as alternativas:
a)
[tex][(\sqrt{10})^2-8]^2=24-8\sqrt{5}[/tex]
[tex](10-8)^2=24-8\sqrt{5}[/tex]
[tex]4=24-8\sqrt{5}[/tex]
Falso.
b)
[tex](4^2-8)^2=24-8\sqrt{5}[/tex]
[tex](16-8)^2=24-8\sqrt{5}[/tex]
[tex]64=24-8\sqrt{5}[/tex]
Falso.
c)
[tex][(2\sqrt{2})^2-8]^2=24-8\sqrt{5}[/tex]
[tex][8-8]^2=24-8\sqrt{5}[/tex]
[tex]0=24-8\sqrt{5}[/tex]
Falso.
d)
[tex][(\sqrt{5}+1)^2-8]^2=24-8\sqrt{5}[/tex]
[tex][1^2+(\sqrt{5})^2+2\sqrt{5}-8]^2=24-8\sqrt{5}[/tex]
[tex](2\sqrt{5}-2)^2=24-8\sqrt{5}[/tex]
[tex](2\sqrt{5})^2+2^2-2\cdot2\cdot2\sqrt{5}=24-8\sqrt{5}[/tex]
[tex]20+4-8\sqrt{5}=24-8\sqrt{5}[/tex]
[tex]24-8\sqrt{5}=24-8\sqrt{5}[/tex]
Verdadeiro.
