Question

How many non zero and positive real numbers x are there such that $x^{x^{\sqrt{x}}}=(x\sqrt{x}{)}^x$?

• A. 1
• B. 2
• C. 4
• D. infinite

Answer 1

The correct option is B 2

Given:

$x^{x^{3/2}}=x^{\frac{3x}{2}}$

Case - I when base $x=1$ then LHS = RHS = 1

Case - II when base $x$ $\ne$ 1

then $x^{3/2}=\frac{3x}{2}$

$\Rightarrow x(\sqrt{x}-\frac{3}{2})=0$

$\Rightarrow x\ne 0\therefore \sqrt{x}=\frac{3}{2}$

$\Rightarrow x=\frac{9}{4}$

Hence two solutions