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Question Number 144760 by mathdanisur last updated on 28/Jun/21

((1+(√x)))^(1/3)  + ((1-(√x)))^(1/3)  = (5)^(1/3)   Find  x=?

$$\sqrt[{\mathrm{3}}]{\mathrm{1}+\sqrt{{x}}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{1}-\sqrt{{x}}}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{5}} \\ $$$${Find}\:\:\boldsymbol{{x}}=? \\ $$

Answered by Olaf_Thorendsen last updated on 28/Jun/21

Let α = ((1+(√x)))^(1/3)  and β = ((1−(√x)))^(1/3)   α+β  = (5)^(1/3)   (α+β)^3   = α^3 +β^3 +3αβ(α+β) = 5  (1+(√x))+(1−(√x))+3(5)^(1/3) ((1−x))^(1/3)  = 5  ((1−x))^(1/3)  = (1/( (5)^(1/3) ))  1−x = (1/5)  x = (4/5)

$$\mathrm{Let}\:\alpha\:=\:\sqrt[{\mathrm{3}}]{\mathrm{1}+\sqrt{{x}}}\:\mathrm{and}\:\beta\:=\:\sqrt[{\mathrm{3}}]{\mathrm{1}−\sqrt{{x}}} \\ $$$$\alpha+\beta\:\:=\:\sqrt[{\mathrm{3}}]{\mathrm{5}} \\ $$$$\left(\alpha+\beta\right)^{\mathrm{3}} \:\:=\:\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\mathrm{3}\alpha\beta\left(\alpha+\beta\right)\:=\:\mathrm{5} \\ $$$$\left(\mathrm{1}+\sqrt{{x}}\right)+\left(\mathrm{1}−\sqrt{{x}}\right)+\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{5}}\sqrt[{\mathrm{3}}]{\mathrm{1}−{x}}\:=\:\mathrm{5} \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{1}−{x}}\:=\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{5}}} \\ $$$$\mathrm{1}−{x}\:=\:\frac{\mathrm{1}}{\mathrm{5}} \\ $$$${x}\:=\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$

Commented by mathdanisur last updated on 29/Jun/21

cool Sir thankyou

$${cool}\:{Sir}\:{thankyou} \\ $$

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