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

 

We can write (1) as 

${\left(\frac{27}{8}\right)}^{1/x}+{\left(\frac{12}{8}\right)}^{1/x}=2$

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

As $3/2>1,(3/2{)}^t>1$ if $t>0$

and $(3/2{)}^t<1$ if  $t<0$.

Thus, ${\left(\frac{3}{2}\right)}^{3/x}+{\left(\frac{3}{2}\right)}^{1/x}<2$ if $x > 0$.

Therefore, (1) is possible if 1/x = 0. But this is not true for any real value of x.