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Question Number 77805 by Dah Solu Tion last updated on 10/Jan/20

$$Solveforx$$$$\frac{8^x+{27}^x}{{12}^x+{18}^x}=\frac{7}{6}$$

Answered by john santu last updated on 10/Jan/20

$${12}^x+{18}^x=3^x\times {\left(2^x\right)}^2+2^x\times {\left(3^x\right)}^2$$$$let{2}^x=t,3^x=y$$$$t^3+y^3=\frac{7}{6}(t^{2}y+{ty}^2)$$$$6(t+y)(t^2-ty+y^2)=7ty(t+y)$$$$6{(t^2-ty+y)}^2=7ty$$$$6t^2-13ty+6y^2=0$$$$(2t-3y)(3t-2y)=0$$$$\left(i\right)2t=3y\Rightarrow 2^{x+1}=3^{x+1}\Rightarrow x=-1$$$$\left(ii\right)3\times 2^x=2\times 3^x\Rightarrow \frac{3}{2}={\left(\frac{3}{2}\right)}^x\Rightarrow x=1$$

Commented by Dah Solu Tion last updated on 10/Jan/20

$$Thankboss$$

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