Question

Number of values of x for which $\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$

• A. 2
• B. 3
• C. 1
• D. No value of x

Answer 1

Answer: (A)

2

Answer 2

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

$\frac{(8)^{x}}{(12)^{x}}\frac{(1 + (27/8)^{x})}{(1 + (18/12)^{x})} = \frac{7}{6}$

⇒$\left( \frac{2}{3} \right)^{x}\left( \frac{1 + (3/2)^{3x}}{1 + (3/2)^{x}} \right) = \frac{7}{6}$

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

⇒ $\frac{1 + t^{3}}{t(1 + t)} = \frac{7}{6}$ Q t + 1 ≠ 0

$\frac{(1 + t)(t^{2} + 1 - t)}{t(1 + t)} = \frac{7}{6}$

⇒ $\frac{t^{2} + 1 - t}{t} = \frac{7}{6}$

⇒ t = $\frac{2}{3}$ or $\frac{3}{2}$