Compute $\sqrt{2}\cdot \sqrt{18}$ and give the final simplified numerical answer.

Expand and fully simplify $\sqrt{80}$ into a product of an integer and a radical.

Rationalize the denominator and simplify: $\frac{5}{\sqrt[3]{2}}$

Simplify $\sqrt{\frac{50x^7}{2x^2}}$.

Select the statement that accurately lists all the conditions required for a radical to be considered fully simplified (assuming principal roots and nonnegative variables).