Question

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

• A. 0
• B. 1
• C. infinite
• D. 2

Answer 1

Answer: (A)

$27^{1 / x}+12^{1 / x}=2 \cdot\left(8^{1 / x}\right)$
$3^{3 / x}+3^{1 / x} \cdot 2^{2 / x}=2 \cdot 2^{3 / x}$
$\left(\frac{3}{2}\right)^{3 / x}+\left(\frac{3}{2}\right)^{1 / x}=2$
assume $\left(\frac{3}{2}\right)^{1 / x }= t$
$ t^3+t-2=0$
$ (t-1)\left(t^2+t+2\right)=0 $
$ \Rightarrow t=1 $
$ \text { so }\left(\frac{3}{2}\right)^{1 / x}=1$
No real solution