Question

Number of soultion(s) of the equation $x^{x\sqrt{x}}=(x\sqrt{x}{)}^x$ is/are:

• A. $0$
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is C $2$

We have $x^{x\sqrt{x}}=(x\sqrt{x}{)}^x$

Take log on both sides

$log\left(x^{x\sqrt{x}}\right)=log(x\sqrt{x}{)}^x$

$⟹x\sqrt{x}log\left(x\right)=xlog\left(x\sqrt{x}\right)$

$⟹x\sqrt{x}log\left(x\right)=xlog\left(x^{\frac{3}{2}}\right)$

$⟹x\sqrt{x}log\left(x\right)=\frac{3x}{2}log\left(x\right)$

$⟹2x\sqrt{x}log\left(x\right)=3xlog\left(x\right)$

$⟹xlog\left(x\right)[2\sqrt{x}-3]=0$

$x$ cannot be $0$ as $0^0$ is indeterminate

$log\left(x\right)=0$ or $2\sqrt{x}-3=0⟹x=1$ or$x=94$

Number of solution of given equation is $2$.