jmc

Since a square root is an exponent of $\frac{1}{2}$ and since exponents distribute across multiplication, we can combine the radicals. $$\sqrt{8}\cdot \sqrt{50}=\sqrt{8\cdot 50}.$$ Now split the radicand into prime factors: $8\cdot 50=2\cdot 2\cdot 2\cdot 2\cdot 5^2=(2\cdot 2{)}^2\cdot 5^2$. We find $\sqrt{8\cdot 50}=\sqrt{(2\cdot 2{)}^2\cdot 5^2}=2\cdot 2\cdot 5=\boxed{20}$.