Question

The number of real solutions of the equation $27^{1/x}+$ $12^{1/x}=2\times 8^{1/x}$ is

• A. one
• B. two
• C. infinite
• D. zero

Answer 1

Answer: (D)

The given equation can be written as
${\left(\frac{3}{2}\right)}^{3/x}+{\left(\frac{3}{2}\right)}^{1/x}=2$
Put ${\left(\frac{3}{2}\right)}^{1/x}=t$, then the equation becomes
$t^3+t-2=0$
$\Rightarrow (t-1)\left(t^2+t+2\right)=0$
But $t^2+t+2=0$ has no real roots,
$\therefore t=1$
$\Rightarrow {\left(\frac{3}{2}\right)}^{1/x}=1$
$\Rightarrow \frac{1}{x}=0$
which is not possible for any value of $x$.