Canceling out $\sqrt{2}$ with $2$

Question

Factor out common term $2$ $$= \frac{2(2+\sqrt{2})}{\sqrt{2}}$$ Cancel the common term $\sqrt{2}$ $$= \sqrt{2}(2+\sqrt{2})$$ canceling out $\sqrt{2}$ with the number $2$, what am I missing? Thank you.

Answer 1

$$\frac{2(2+\sqrt2)}{\sqrt2}=\color{red}{\frac2{\sqrt2}}(2+\sqrt2)=\color{red}{\sqrt2}(2+\sqrt2)=2\sqrt2+2$$

Answer 2

$$\frac{2}{\sqrt{2}}=\frac{2}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{2\times\sqrt{2}}{2}=\frac{2}{2}\times\sqrt{2}=\sqrt{2}$$