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Q.
The number of real solutions of the equation $27^{1 / x}+$ $12^{1 / x}=2 \times 8^{1 / x}$ is
Complex Numbers and Quadratic Equations
Solution:
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 $x$.