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