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Question Number 160831 by MathsFan last updated on 07/Dec/21

simplify  ((2+(√5)))^(1/3)

$$\mathrm{simplify}\:\:\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}} \\ $$

Commented by mr W last updated on 07/Dec/21

((2+(√5)))^(1/3)   =(((16+8(√5)))^(1/3) /2)  =(((5(√5)+3(√5)+3×5+1))^(1/3) /2)  =(((((√5))^3 +3((√5))^2 +3(√5)+1))^(1/3) /2)  =(((((√5)+1)^3 ))^(1/3) /2)  =(((√5)+1)/2)=φ=golden ratio

$$\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}} \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\mathrm{16}+\mathrm{8}\sqrt{\mathrm{5}}}}{\mathrm{2}} \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\mathrm{5}\sqrt{\mathrm{5}}+\mathrm{3}\sqrt{\mathrm{5}}+\mathrm{3}×\mathrm{5}+\mathrm{1}}}{\mathrm{2}} \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{5}}\right)^{\mathrm{3}} +\mathrm{3}\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2}} +\mathrm{3}\sqrt{\mathrm{5}}+\mathrm{1}}}{\mathrm{2}} \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{5}}+\mathrm{1}\right)^{\mathrm{3}} }}{\mathrm{2}} \\ $$$$=\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{2}}=\phi={golden}\:{ratio} \\ $$

Commented by MathsFan last updated on 08/Dec/21

thank you sir  but is there any formula for   questions like this please?

$${thank}\:{you}\:{sir} \\ $$$${but}\:{is}\:{there}\:{any}\:{formula}\:{for}\: \\ $$$${questions}\:{like}\:{this}\:{please}? \\ $$

Commented by mr W last updated on 08/Dec/21

no! one knows it or one doesn′t know  it. there is no general way to solve it  through a formula!

$${no}!\:{one}\:{knows}\:{it}\:{or}\:{one}\:{doesn}'{t}\:{know} \\ $$$${it}.\:{there}\:{is}\:{no}\:{general}\:{way}\:{to}\:{solve}\:{it} \\ $$$${through}\:{a}\:{formula}! \\ $$

Commented by MathsFan last updated on 08/Dec/21

okay Sir.  Thank you once again

$${okay}\:{Sir}. \\ $$$${Thank}\:{you}\:{once}\:{again} \\ $$

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