Question

Me ajudem por favorrr

Answer 1

Resposta:

d)

Explicação passo-a-passo:

Considerando que $a+b=k$:

$k^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2$

$k^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}\right)^2+\left(\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2+2\cdot\sqrt{4+\sqrt{10+2\sqrt{5}}}\cdot\sqrt{4-\sqrt{10+2\sqrt{5}}}$

$k^2=4+\sqrt{10+2\sqrt{5}}}+4-\sqrt{10+2\sqrt{5}}+2\cdot\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\cdot\left(4-\sqrt{10+2\sqrt{5}}\right)}$

$k^2=8+2\cdot\sqrt{4^2-\left(\sqrt{10+2\sqrt{5}}\right)^2}$

$k^2=8+2\cdot\sqrt{16-(10+2\sqrt{5})}$

$k^2=8+2\cdot\sqrt{6-2\sqrt{5}}$

$k^2-8=2\cdot\sqrt{6-2\sqrt{5}}$

$(k^2-8)^2=\left(2\cdot\sqrt{6-2\sqrt{5}}\right)^2$

$(k^2-8)^2=4\cdot(6-2\sqrt{5})$

$(k^2-8)^2=24-8\sqrt{5}$

Vamos agora testar as alternativas:

a)

$[(\sqrt{10})^2-8]^2=24-8\sqrt{5}$

$(10-8)^2=24-8\sqrt{5}$

$4=24-8\sqrt{5}$

Falso.

b)

$(4^2-8)^2=24-8\sqrt{5}$

$(16-8)^2=24-8\sqrt{5}$

$64=24-8\sqrt{5}$

Falso.

c)

$[(2\sqrt{2})^2-8]^2=24-8\sqrt{5}$

$[8-8]^2=24-8\sqrt{5}$

$0=24-8\sqrt{5}$

Falso.

d)

$[(\sqrt{5}+1)^2-8]^2=24-8\sqrt{5}$

$[1^2+(\sqrt{5})^2+2\sqrt{5}-8]^2=24-8\sqrt{5}$

$(2\sqrt{5}-2)^2=24-8\sqrt{5}$

$(2\sqrt{5})^2+2^2-2\cdot2\cdot2\sqrt{5}=24-8\sqrt{5}$

$20+4-8\sqrt{5}=24-8\sqrt{5}$

$24-8\sqrt{5}=24-8\sqrt{5}$

Verdadeiro.