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Question Number 213791 by hardmath last updated on 16/Nov/24

$$\mathrm{If}\mathrm{x}-\sqrt[3]{\mathrm{x}}-\frac{4}{\sqrt[3]{\mathrm{x}}}=10$$$$\mathrm{Find}\sqrt[3]{\mathrm{x}}-\frac{1}{\sqrt[3]{\mathrm{x}}}+3=?$$

Commented by muallimRiyoziyot last updated on 19/Nov/24

$$x-8=\sqrt[3]{x}+2+\frac{4}{\sqrt[3]{x}}$$$$(\sqrt[3]{x}-2)(\sqrt[3]{x^2}+2\sqrt[3]{x}+4)=\frac{\sqrt[3]{x^2}+2\sqrt[3]{x}+4}{\sqrt[3]{x}}$$$$\sqrt[3]{x}-2=\frac{1}{\sqrt[3]{x}}\to \sqrt[3]{x}-\frac{1}{\sqrt[3]{x}}=2$$

Commented by muallimRiyoziyot last updated on 19/Nov/24

$$2+3=5$$

Answered by Ghisom last updated on 17/Nov/24

$$x^{4/3}-x^{2/3}-10x^{1/3}-4=0$$$$(x^{2/3}-2x^{1/3}-1)(x^{2/3}+2x^{1/3}+4)=0$$$$$$$$x^{1/3}-\frac{1}{x^{1/3}}+3=y$$$$x^{2/3}-(y-3)x^{1/3}-1=0$$$$y-3=2\Rightarrow y=5$$

Answered by a.lgnaoui last updated on 17/Nov/24

$$\mathrm{posons}\mathbf{z}{=}^3\sqrt{\mathbf{x}}$$$${\mathbf{z}}^3-\mathbf{z}-\frac{4}{\mathbf{z}}=10$$$${\mathbf{z}}^4-{\mathbf{z}}^2-10\mathbf{z}-4=0\left(1\right)$$$${\mathbf{z}}^2+3\mathbf{z}-1=?$$$$\left(1\right){[{({\mathbf{z}}^2-\frac{1}{2})}^2-\frac{17}{4}]}^2=100({\mathbf{z}}^2-\frac{1}{2})+50$$$${({\mathbf{k}}^2-\frac{17}{4})}^2=100{\mathbf{k}}^2+50(\mathbf{X}={\mathbf{z}}^2-\frac{1}{2})$$$${\mathbf{X}}^4-\frac{17}{2}{\mathbf{X}}^2-100{\mathbf{X}}^2+\frac{{17}^2}{4^2}-50=0$$$${\mathbf{k}}^4-\left(\frac{217}{2}\right)\mathbf{k}-\frac{511}{4^2}=0$$$${217}^2+511/64=\frac{{217}^2\times 4+511}{64}$$$$108,647$$$$\mathbf{X}=\frac{217}{4}+\frac{108.647}{2}=\frac{217.294+217}{4}$$$$=\frac{434.294}{4}=108.573$$$${\mathrm{z}}^2=109.073\mathrm{z}=10,443$$$$3\sqrt{\mathrm{x}}=10,443$$$$$$$${}^3\sqrt{\mathrm{x}}-1{/}^3\sqrt{\mathrm{x}}+3=5,19$$$$$$$$$$

Commented by Ghisom last updated on 17/Nov/24

$$\mathrm{this}\mathrm{is}\mathrm{wrong}$$

Commented by a.lgnaoui last updated on 17/Nov/24

$$\sqrt{\mathrm{x}}=\frac{10,443}{3}$$$$\sqrt{\mathrm{x}}-\frac{1}{\sqrt{\mathrm{x}}}+3=5,193(\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{unique}$$$$\mathrm{solution})$$

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