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x-3-1-x-3-18-3-Find-the-value-of-x-

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Question Number 224034 by MirHasibulHossain last updated on 15/Aug/25
x^3 +(1/x^3 )=18(√3) .Find the value of x.
$$\mathrm{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }=\mathrm{18}\sqrt{\mathrm{3}}\:.\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$
Answered by BaliramKumar last updated on 15/Aug/25
x = ((11(√2) +9(√3) ))^(1/3)
$${x}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{11}\sqrt{\mathrm{2}}\:+\mathrm{9}\sqrt{\mathrm{3}}\:\:} \\ $$
Answered by Ghisom last updated on 15/Aug/25
(x^3 )^2 −18(√3)x^3 +1=0 x^3 =9(√3)±11(√2) x^3 =((√3)±(√2))^3 x=((√3)±(√2))ω^k [ω=−(1/2)+((√3)/2)i∧k=0, 1, 2]
$$\left({x}^{\mathrm{3}} \right)^{\mathrm{2}} −\mathrm{18}\sqrt{\mathrm{3}}{x}^{\mathrm{3}} +\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{3}} =\mathrm{9}\sqrt{\mathrm{3}}\pm\mathrm{11}\sqrt{\mathrm{2}} \\ $$$${x}^{\mathrm{3}} =\left(\sqrt{\mathrm{3}}\pm\sqrt{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$${x}=\left(\sqrt{\mathrm{3}}\pm\sqrt{\mathrm{2}}\right)\omega^{{k}} \\ $$$$\:\:\:\:\:\left[\omega=−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}\wedge{k}=\mathrm{0},\:\mathrm{1},\:\mathrm{2}\right] \\ $$
Commented by RedstoneGG4 last updated on 17/Aug/25
Impressive simplification of 9(√3)±11(√2).
$$\mathrm{Impressive}\:\mathrm{simplification}\:\mathrm{of}\:\mathrm{9}\sqrt{\mathrm{3}}\pm\mathrm{11}\sqrt{\mathrm{2}}. \\ $$
Answered by RedstoneGG4 last updated on 17/Aug/25
x^6 + 1 = 18(√3)x^3 : x ≠ 0 x^6 − 18(√3)x^3 + 1 = 0 x^6 − 18(√3)x^3 +(9(√3))^2 = (9(√3))^2 − 1 (x^3 − 9(√3))^2 = 242 x^3 − 9(√3) = ±11(√2) x^3 = 9(√3) ± 11(√2) x ∈ R ⇔ x = ((9(√3) ± 11(√2)))^(1/3) ∈ {∽0.3178, ∽3.1462} x ∈ C ⇔ k ∈ Z: x = (((9(√3) ± 11(√2))e^(2πki) ))^(1/3) → x ∈ {((9(√3) + 11(√2)))^(1/3) , ((9(√3) − 11(√2)))^(1/3) , -(1/2)((9(√3) + 11(√2)))^(1/3) + ((√3)/2)((9(√3) + 11(√2)))^(1/3) i, -(1/2)((9(√3) − 11(√2)))^(1/3) + ((√3)/2)((9(√3) − 11(√2)))^(1/3) i, -(1/2)((9(√3) + 11(√2)))^(1/3) − ((√3)/2)((9(√3) + 11(√2)))^(1/3) i, -(1/2)((9(√3) − 11(√2)))^(1/3) − ((√3)/2)((9(√3) − 11(√2)))^(1/3) i}
$${x}^{\mathrm{6}} \:+\:\mathrm{1}\:=\:\mathrm{18}\sqrt{\mathrm{3}}{x}^{\mathrm{3}} :\:{x}\:\neq\:\mathrm{0} \\ $$$${x}^{\mathrm{6}} \:−\:\mathrm{18}\sqrt{\mathrm{3}}{x}^{\mathrm{3}} \:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$${x}^{\mathrm{6}} \:−\:\mathrm{18}\sqrt{\mathrm{3}}{x}^{\mathrm{3}} \:+\left(\mathrm{9}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:=\:\left(\mathrm{9}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:−\:\mathrm{1} \\ $$$$\left({x}^{\mathrm{3}} \:−\:\mathrm{9}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:=\:\mathrm{242} \\ $$$${x}^{\mathrm{3}} \:−\:\mathrm{9}\sqrt{\mathrm{3}}\:=\:\pm\mathrm{11}\sqrt{\mathrm{2}} \\ $$$${x}^{\mathrm{3}} \:=\:\mathrm{9}\sqrt{\mathrm{3}}\:\pm\:\mathrm{11}\sqrt{\mathrm{2}} \\ $$$${x}\:\in\:\mathbb{R}\:\Leftrightarrow\:{x}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:\pm\:\mathrm{11}\sqrt{\mathrm{2}}}\:\in\:\left\{\backsim\mathrm{0}.\mathrm{3178},\:\backsim\mathrm{3}.\mathrm{1462}\right\} \\ $$$${x}\:\in\:\mathbb{C}\:\Leftrightarrow\:{k}\:\in\:\mathbb{Z}:\:{x}\:=\:\sqrt[{\mathrm{3}}]{\left(\mathrm{9}\sqrt{\mathrm{3}}\:\pm\:\mathrm{11}\sqrt{\mathrm{2}}\right){e}^{\mathrm{2}\pi{ki}} }\:\rightarrow \\ $$$${x}\:\in\:\left\{\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:+\:\mathrm{11}\sqrt{\mathrm{2}}},\:\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:−\:\mathrm{11}\sqrt{\mathrm{2}}},\:-\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:+\:\mathrm{11}\sqrt{\mathrm{2}}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:+\:\mathrm{11}\sqrt{\mathrm{2}}}{i},\:-\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:−\:\mathrm{11}\sqrt{\mathrm{2}}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:−\:\mathrm{11}\sqrt{\mathrm{2}}}{i},\:-\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:+\:\mathrm{11}\sqrt{\mathrm{2}}}\:−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:+\:\mathrm{11}\sqrt{\mathrm{2}}}{i},\:-\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:−\:\mathrm{11}\sqrt{\mathrm{2}}}\:−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{9}\sqrt{\mathrm{3}}\:−\:\mathrm{11}\sqrt{\mathrm{2}}}{i}\right\} \\ $$$$ \\ $$
Commented by RedstoneGG4 last updated on 17/Aug/25
Correction: x ∈ R ⇔ x = (√3) ± (√2) x ∈ C ⇔ x ∈ {(√3) + (√2), (√3) − (√2), -(((√3) + (√2))/2) + ((3 + (√6))/2)i, -(((√3) + (√2))/2) − ((3 + (√6))/2), (((√2) − (√3))/2) + ((3 − (√6))/2)i, (((√2) − (√3))/2) + (((√6) − 3)/2)i}
$$\mathrm{Correction}: \\ $$$${x}\:\in\:\mathbb{R}\:\Leftrightarrow\:{x}\:=\:\sqrt{\mathrm{3}}\:\pm\:\sqrt{\mathrm{2}} \\ $$$${x}\:\in\:\mathbb{C}\:\Leftrightarrow\:{x}\:\in\:\left\{\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}},\:\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}},\:-\frac{\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}}{\mathrm{2}}\:+\:\frac{\mathrm{3}\:+\:\sqrt{\mathrm{6}}}{\mathrm{2}}{i},\:-\frac{\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}}{\mathrm{2}}\:−\:\frac{\mathrm{3}\:+\:\sqrt{\mathrm{6}}}{\mathrm{2}},\:\frac{\sqrt{\mathrm{2}}\:−\:\sqrt{\mathrm{3}}}{\mathrm{2}}\:+\:\frac{\mathrm{3}\:−\:\sqrt{\mathrm{6}}}{\mathrm{2}}{i},\:\frac{\sqrt{\mathrm{2}}\:−\:\sqrt{\mathrm{3}}}{\mathrm{2}}\:+\:\frac{\sqrt{\mathrm{6}}\:−\:\mathrm{3}}{\mathrm{2}}{i}\right\} \\ $$

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