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Question Number 214443 by hardmath last updated on 08/Dec/24

$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\in \mathrm{Q}$$$$\frac{1}{\sqrt{2}-\sqrt[3]{2}}=2^{\mathbf{a}}+2^{\mathbf{b}}+2^{\mathbf{c}}+2^{\mathbf{d}}+2^{\mathbf{e}}+2^{\mathbf{f}}$$$$\mathrm{find}:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}=?$$

Answered by ajfour last updated on 08/Dec/24

$$\frac{1}{a^3-a^2}=\frac{a^6+a^4+a^5}{a^9-a^6}=\frac{2+2^{2/3}+2^{5/6}}{2\sqrt{2}-2}$$$$=\frac{(2+2^{2/3}+2^{5/6})(2\sqrt{2}+2)}{4}$$$$=(1+\frac{1}{2^{1/3}}+\frac{1}{2^{1/6}})(2^{1/2}+1)$$$$=1+\frac{1}{2^{1/3}}+\frac{1}{2^{1/6}}+2^{1/2}+2^{1/6}+2^{1/3}$$$$willthisdo?$$

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