# Aptitude Question ID : 93836

If $x^{\sqrt[x]{x}} = \left ( x\sqrt{x} \right )^{x}$, then x equals :
[A]$\frac{3}{2}$
[B]$\frac{9}{4}$
[C]$\frac{4}{9}$
[D]$\frac{2}{3}$

$\mathbf{\frac{9}{4}}$
Given Expression,
$x^{\sqrt[x]{x}} = \left ( x\sqrt{x} \right )^{x}$
$=> x^{x.x^{\frac{1}{2}}} = \left ( x\times x^{\frac{1}{2}}\right )^{x}$
$=> x^{x^{1+\frac{1}{2}}} = \left ( x^{1+\frac{1}{2}} \right )^{x}$
$=> x^{x^{\frac{3}{2}}} = \left ( x\frac{3}{2} \right )^{x} = x^{\frac{3x}{2}}$
$=> x^{\frac{3x}{2}} = \frac{3x}{2}= x^{\frac{3x}{2}} - \frac{3x}{2} = 0$
$=> x\left ( x^{\frac{1}{2}} - \frac{3}{2} \right ) = 0$
$=> x = 0$
or $x^{\frac{1}{2}} = \frac{3}{2}$
$=> x = \left ( \frac{3}{2} \right )^{2} = \frac{9}{4}$
x = 0 given indeterminate value.
$\therefore x = \frac{9}{4}$
Hence option [B] is the right answer.