Question

If $x^{x\sqrt{x}}=(x\sqrt{x}{)}^x$, then x equals

• A. $\frac{4}{9}$
• B. $\frac{2}{3}$
• C. $\frac{3}{2}$
• D. $\frac{9}{4}$

Answer 1

The correct option is D $\frac{9}{4}$

$x^{x\sqrt{x}}=(x\sqrt{x}{)}^x$
$=x^x\cdot x^{\left(\frac{1}{\sqrt{2}}\right)\cdot x}$
$x^{x\cdot x\left(\frac{1}{2}\right)}=x^x\cdot x^{\frac{x}{2}}$
$x^{x\left(\frac{3}{2}\right)}=x^{\left(\frac{3}{2}\right)x}$
$x^{\frac{3}{2}}=\frac{3}{2}x$
$\therefore x\sqrt{x}=\frac{3}{2}x⟹\sqrt{x}=\frac{3}{2}⟹x=\frac{9}{4}$