First of all, use the fact that #\sqrt{a\cdot b}=\sqrt{a}\sqrt{b}#. So, we have that #\sqrt{x^3y^9}=\sqrt{x^3}\sqrt{y^9}#

Let's deal with a root at a time: we can write #x^3# as #x^2\cdot x#. So, #\sqrt{x^3}=\sqrt{x^2\cdot x}=\sqrt{x^2}\sqrt{x}=x\sqrt{x}#.

For the same reasons, we write #y^9# as #y^2\cdoty^2\cdoty^2\cdoty^2\cdot y#. So, we have that #\sqrt{y^9}=\sqrt{y^2\cdoty^2\cdoty^2\cdoty^2\cdot y}=\sqrt{y^2}\sqrt{y^2}\sqrt{y^2}\sqrt{y^2}\sqrt{y}#, which equals #y^4\sqrt{y}#.

Putting the two things together, we have that

#\sqrt{x^3y^9} = xy^4\sqrt{xy}#.