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Question Number 123845 by Akeyz last updated on 28/Nov/20

Commented by TANMAY PANACEA last updated on 28/Nov/20

$$n=1$$$$\frac{8+27}{12+18}=\frac{35}{30}=\frac{7}{6}$$

Answered by floor(10²Eta[1]) last updated on 28/Nov/20

$$6.(2^{3\mathrm{n}}+3^{3\mathrm{n}})=7(2^{2\mathrm{n}}.3^{\mathrm{n}}+2^{\mathrm{n}}.3^{2\mathrm{n}})$$$$2^{\mathrm{n}}=\mathrm{a},3^{\mathrm{n}}=\mathrm{b}$$$$6({\mathrm{a}}^3+{\mathrm{b}}^3)=7({\mathrm{a}}^2\mathrm{b}+{\mathrm{ab}}^2)$$$$6(\mathrm{a}+\mathrm{b})({\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2)=7\mathrm{ab}(\mathrm{a}+\mathrm{b})$$$$6({\mathrm{a}}^2+{\mathrm{b}}^2-\mathrm{ab})=7\mathrm{ab}$$$${6\mathrm{a}}^2-13\mathrm{ab}+{6\mathrm{b}}^2=0$$$$\mathrm{a}=\frac{13\mathrm{b}\pm 5\mathrm{b}}{12}=\frac{3\mathrm{b}}{2},\frac{2\mathrm{b}}{3}$$$$2^{\mathrm{n}}=\frac{3.3^{\mathrm{n}}}{2}\Rightarrow 2^{\mathrm{n}+1}=3^{\mathrm{n}+1}\Rightarrow \mathrm{n}=-1$$$$2^{\mathrm{n}}=\frac{2.3^{\mathrm{n}}}{3}\Rightarrow 2^{\mathrm{n}-1}=3^{\mathrm{n}-1}\Rightarrow \mathrm{n}=1$$

Commented by WilliamsErinfolami last updated on 28/Nov/20

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