Question

$x^{x\sqrt{x}}=(x\sqrt{x}{)}^x$. Find the value of x

Answer 1

$x^{x\sqrt{x}}=\left(x\sqrt{x}{)}^x\right(x^x{)}^{\sqrt{x}}=x^x\left(\sqrt{x}{)}^x\right(x^x{)}^{\sqrt{x}-1}=\left(\sqrt{x}{)}^x\right((\sqrt{x}{)}^{2x}{)}^{\sqrt{x}-1}=(\sqrt{x}{)}^x\Rightarrow 2\sqrt{x}(\sqrt{x}-1)=x2\sqrt{x}-2=12\sqrt{x}=3\Rightarrow x=\frac{9}{4}$