how do rationalize and simplify 2x/sqrt(8x)

how do rationalize and simplify 2x/sqrt(8x)

$$\dfrac{2x}{ \sqrt{8x} } = \sqrt{ \dfrac{4x^2}{ 8x } } = \sqrt{ \dfrac{x}{2} } = \frac{1}{2}\sqrt{ 2x }$$

$$\frac{2x}{\sqrt{8x}}=\frac{2x}{\sqrt{8x}}*\frac{\sqrt{2}}{\sqrt{2}}=\frac{2x\sqrt{2}}{\sqrt{16x}}=\frac{2x\sqrt{2}}{\sqrt{16}*\sqrt{x}}=\frac{2x\sqrt{2}}{4\sqrt{x}}=\frac{2\sqrt{x}\sqrt{2}}{4}=\frac{1}{2}\sqrt{2x}$$