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Question Number 182188 by mathlove last updated on 05/Dec/22

((((√5)+2))^(1/3) +(((√5)−2))^(1/3) )^(2014) =?

$$\left(\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}+\mathrm{2}}+\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}−\mathrm{2}}\right)^{\mathrm{2014}} =? \\ $$

Answered by Frix last updated on 05/Dec/22

((2+(√5)))^(1/3) =ϕ∧((−2+(√5)))^(1/3) =(1/ϕ)  ϕ+(1/ϕ)=(√5)  ((√5))^(2014) =5^(1007)

$$\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}}=\varphi\wedge\sqrt[{\mathrm{3}}]{−\mathrm{2}+\sqrt{\mathrm{5}}}=\frac{\mathrm{1}}{\varphi} \\ $$$$\varphi+\frac{\mathrm{1}}{\varphi}=\sqrt{\mathrm{5}} \\ $$$$\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2014}} =\mathrm{5}^{\mathrm{1007}} \\ $$

Answered by mnjuly1970 last updated on 05/Dec/22

   x^3 = 2(√5) + 3 x ⇒ x =(√5)      answer = 5^( 1007)

$$\:\:\:{x}^{\mathrm{3}} =\:\mathrm{2}\sqrt{\mathrm{5}}\:+\:\mathrm{3}\:{x}\:\Rightarrow\:{x}\:=\sqrt{\mathrm{5}} \\ $$$$\:\:\:\:{answer}\:=\:\mathrm{5}^{\:\mathrm{1007}} \\ $$

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