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Question Number 132285 by bemath last updated on 13/Feb/21

$$\mathrm{Simplify}\mathrm{the}\mathrm{equation}\mathrm{of}$$$$\frac{({\mathrm{x}}^{\frac{1}{3}}-{\mathrm{x}}^{\frac{1}{6}})({\mathrm{x}}^{\frac{1}{2}}+\mathrm{x})({\mathrm{x}}^{\frac{1}{2}}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}{({\mathrm{x}}^{\frac{4}{3}}-\mathrm{x})(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$$$\mathrm{with}\mathrm{x}\ne 0$$$$$$

Answered by Olaf last updated on 13/Feb/21

$$\mathrm{X}=\frac{({\mathrm{x}}^{\frac{1}{3}}-{\mathrm{x}}^{\frac{1}{6}})({\mathrm{x}}^{\frac{1}{2}}+\mathrm{x})({\mathrm{x}}^{\frac{1}{2}}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}{({\mathrm{x}}^{\frac{4}{3}}-\mathrm{x})(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$$$\mathrm{Let}a=x^{\frac{1}{6}}$$$$\mathrm{X}=\frac{(a^2-a)(a^3+a^6)(a^3+a^2+a^4)}{(a^8-a)(a^6+a^2+a^4)}$$$$\mathrm{X}=\frac{a(a-1)a^3(a^3+1)a^2(a^2+a+1)}{a(a^7-1)a^2(a^4+a^2+1)}$$$$\mathrm{X}=\frac{a^3(a-1)(a^3+1)(a^2+a+1)}{(a^7-1)(a^4+a^2+1)}$$$$\mathrm{X}=\frac{a^3(a-1)(a+1)(a^2-a+1)(a^2+a+1)}{(a^7-1)(a^4+a^2+1)}$$$$\mathrm{X}=\frac{a^3(a-1)(a+1)(a^4+a^2+1)}{(a^7-1)(a^4+a^2+1)}$$$$\mathrm{X}=\frac{a^3(a-1)(a+1)}{a^7-1}=\frac{a^3(a^2-1)}{a^7-1}$$$$\mathrm{X}=\frac{x^{\frac{1}{2}}(x^{\frac{1}{3}}-1)}{x^{\frac{7}{6}}-1}$$$$...\mathrm{or}\mathrm{X}=\frac{a^3(a+1)}{a^6+a^5+a^4+a^3+a^2+a+1}$$

Answered by benjo_mathlover last updated on 13/Feb/21

$$\frac{{\mathrm{x}}^{\frac{1}{6}}({\mathrm{x}}^{\frac{1}{6}}-1){\mathrm{x}}^{\frac{1}{2}}(1+{\mathrm{x}}^{\frac{1}{2}})({\mathrm{x}}^{\frac{1}{2}}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}{\mathrm{x}({\mathrm{x}}^{\frac{1}{3}}-1)(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$$$=\frac{({\mathrm{x}}^{\frac{1}{6}}-1)(1+{\mathrm{x}}^{\frac{1}{2}})({\mathrm{x}}^{\frac{1}{2}}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}{{\mathrm{x}}^{\frac{1}{3}}({\mathrm{x}}^{\frac{1}{6}}-1)({\mathrm{x}}^{\frac{1}{6}}+1)(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$$$=\frac{(1+{\mathrm{x}}^{\frac{1}{2}})({\mathrm{x}}^{\frac{1}{2}}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}{{\mathrm{x}}^{\frac{1}{3}}({\mathrm{x}}^{\frac{1}{6}}+1)(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$$$=\frac{(1+{\mathrm{x}}^{\frac{1}{2}})({\mathrm{x}}^{\frac{1}{6}}+1+{\mathrm{x}}^{\frac{1}{3}})}{({\mathrm{x}}^{\frac{1}{6}}+1)(\mathrm{x}+{\mathrm{x}}^{\frac{1}{3}}+{\mathrm{x}}^{\frac{2}{3}})}$$

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