Solve-for-x-8-x-27-x-12-x-18-x-7-6-

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

S o l v e f o r x

\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 {(2^{x})}^{2} + 2^{x} \times {(3^{x})}^{2}

l e t 2^{x} = t, 3^{x} = y

t^{3} + y^{3} = \frac{7}{6} (t^{2} y + {t y}^{2})

6 (t + y) (t^{2} - t y + y^{2}) = 7 t y (t + y)

6 {(t^{2} - t y + y)}^{2} = 7 t y

6 t^{2} - 13 t y + 6 y^{2} = 0

(2 t - 3 y) (3 t - 2 y) = 0

(i) 2 t = 3 y \Rightarrow 2^{x + 1} = 3^{x + 1} \Rightarrow x = - 1

(i i) 3 \times 2^{x} = 2 \times 3^{x} \Rightarrow \frac{3}{2} = {(\frac{3}{2})}^{x} \Rightarrow x = 1

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

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