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Question Number 122377 by aristarque last updated on 16/Nov/20

Commented by Dwaipayan Shikari last updated on 16/Nov/20

$$\sqrt[3]{54\sqrt{3}+41\sqrt{5}}=\sqrt[3]{{(2\sqrt{3}+\sqrt{5})}^3}$$$$\sqrt[3]{54\sqrt{3}-41\sqrt{5}}=\sqrt{{(2\sqrt{3}-\sqrt{5})}^3}$$$$A=\frac{2\sqrt{3}+\sqrt{5}+2\sqrt{3}-\sqrt{5}}{\sqrt{3}}=4$$

Commented by Dwaipayan Shikari last updated on 16/Nov/20

$$Generally$$$$\sqrt[3]{a\sqrt{b}+c\sqrt{d}}=x\sqrt{y}+p\sqrt{z}$$$$a\sqrt{b}+c\sqrt{d}=x^{3}y\sqrt{y}+p^{3}z\sqrt{z}+3x^{2}yp\sqrt{z}+3p^{2}zx\sqrt{y}$$$$54\sqrt{3}+41\sqrt{5}=(x^{3}y+3p^{2}zx)\sqrt{y}+41(p^{3}z+3x^{2}yp)\sqrt{z}$$$$z=5$$$$y=3x^{3}y+3p^{2}zx=54\Rightarrow 3x^3+15p^{2}x=54\Rightarrow 3x(x^2+5p^2)=54$$$$$$$$p^{3}z+3x^{2}yp=41$$$$$$$$5p^3+9x^{2}p=41\Rightarrow p(5p^2+9x^2)=41$$$$$$$$Sox=2$$$$p=1\left(Onepair\right)$$$$$$$$x\sqrt{y}+p\sqrt{z}=2\sqrt{3}+\sqrt{5}$$

Commented by Dwaipayan Shikari last updated on 16/Nov/20

$$Onwhichpart?$$

Commented by aristarque last updated on 16/Nov/20

$$pleaseido{\mathit{n}}^{.}\mathit{t}\mathit{u}\mathit{n}\mathit{d}\mathit{e}\mathit{r}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{d}\mathit{t}\mathit{h}\mathit{i}\mathit{s}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{p}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{t}\mathit{a}\mathit{i}\mathit{l}$$

Answered by sewak last updated on 17/Nov/20

$$\mathrm{A}=\frac{\sqrt[3]{54\sqrt{3}+41\sqrt{5}}}{\sqrt{3}}+\frac{\sqrt[3]{54\sqrt{3}-41\sqrt{5}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{54\sqrt{3}+41\sqrt{5}}+\sqrt[3]{54\sqrt{3}-41\sqrt{5}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{54\sqrt{3}+41\sqrt{5}+54\sqrt{3}-41\sqrt{5}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{54\sqrt{3}+54\sqrt{3}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{2\times 54\sqrt{3}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{108\sqrt{3}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{\sqrt[3]{3\times 3\times 3\times 4\sqrt{3}}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{3\sqrt[3]{4\sqrt{3}}}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}$$$$\mathrm{or},\mathrm{A}=\frac{3\sqrt[3]{4\sqrt{3}}\times \sqrt{3}}{3}$$$$\mathrm{or},\mathrm{A}=\sqrt[3]{2^2\sqrt{3}}\times \sqrt{3}$$$$\mathrm{or},\mathrm{A}=\sqrt[3]{\sqrt{6}}\times \sqrt{3}$$$$\mathrm{or},\mathrm{A}=\sqrt[6]{6}\times \sqrt{3}$$$$\mathrm{or},\mathrm{A}=1.34\times 1.73=2.3$$

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