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Question Number 99094 by bemath last updated on 18/Jun/20
find ((9+9((9+9((9+9((9+...))^(1/(3  )) ))^(1/(3  )) ))^(1/(3  )) ))^(1/(3  )) −  (√(8−(√(8−(√(8+(√(8−(√(8−(√(8−(√(8−(√)))))))...))))))))
$$\mathrm{find}\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9}\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+…}}}}− \\ $$$$\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}+\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{}}}}…}}}}\: \\ $$
Answered by bobhans last updated on 18/Jun/20
y = ((9+9y))^(1/(3  ))  ⇒y^3 −9y−9 = 0  x=(√(8−(√(8−(√(8+x)))))) ⇒ x^3 +3x^2 −6x−17 = 0  ⇒(x+1)^3 −9(x+1)−9 = 0  let y = x+1 ⇒(x+1)^3 −9(x+1)−9   then (x+1)^3 −9(x+1)−9 −((x+1)^3 −9(x+1)−9 ) = 0
$$\mathrm{y}\:=\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{9}+\mathrm{9y}}\:\Rightarrow\mathrm{y}^{\mathrm{3}} −\mathrm{9y}−\mathrm{9}\:=\:\mathrm{0} \\ $$$$\mathrm{x}=\sqrt{\mathrm{8}−\sqrt{\mathrm{8}−\sqrt{\mathrm{8}+\mathrm{x}}}}\:\Rightarrow\:\mathrm{x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} −\mathrm{6x}−\mathrm{17}\:=\:\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{9}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{9}\:=\:\mathrm{0} \\ $$$$\mathrm{let}\:\mathrm{y}\:=\:\mathrm{x}+\mathrm{1}\:\Rightarrow\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{9}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{9}\: \\ $$$$\mathrm{then}\:\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{9}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{9}\:−\left(\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{9}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{9}\:\right)\:=\:\mathrm{0} \\ $$

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