Question

The number of real solutions of the equation ${27}^{\frac{1}{x}}+{12}^{\frac{1}{x}}=2\left(8^{\frac{1}{x}}\right)$ is

• A. One
• B. Two
• C. Infinite
• D. Zero

Answer 1

The correct option is D

Zero

The given equation can be written as

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

put ${\left(\frac{3}{2}\right)}^{\frac{1}{x}}=t$, then the equation becomes $t^3+t-2=0$

$\Rightarrow (t-1)(t^2+t+2)=0$

But $t^2+t+2=0$ has no real roots, $\therefore t=1$

$\Rightarrow {\left(\frac{3}{2}\right)}^{\frac{1}{x}}=1\Rightarrow \frac{1}{x}=0$