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Math Help - Expo-equation

  1. #1
    Super Member dhiab's Avatar
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    Expo-equation

    Solve in R : \frac{{8^x + 27^x }}{{12^x + 18^x }} = \frac{7}{6}<br />
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  2. #2
    MHF Contributor red_dog's Avatar
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    \frac{2^{3x}+3^{3x}}{2^{2x}3^x+2^x3^{2x}}=\frac{7}  {6}

    \frac{(2^x+3^x)(2^{2x}-2^x3^x+3^{2x})}{2^x3^x(2^x+3^x)}=\frac{7}{6}

    \frac{2^{2x}-2^x3^x+3^{2x}}{2^x3^x}=\frac{7}{6}

    6\cdot 2^{2x}-132^x3^x+6\cdot 3^{2x}=0

    Divide by 3^{2x}:

    6\left(\frac{2}{3}\right)^{2x}-13\left(\frac{2}{3}\right)^x+6=0

    Let \left(\frac{2}{3}\right)^x=t. Then

    6t^2-13t+6=0\Rightarrow t_1=\frac{2}{3}, \ t_2=\frac{3}{2}

    \left(\frac{2}{3}\right)^x=\frac{2}{3}\Rightarrow x=1

    \left(\frac{2}{3}\right)^x=\frac{3}{2}\Rightarrow x=-1
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