Homework Help: What is a laplacian of a laplacian?

1. The problem statement, all variables and given/known data
What is a laplacian of a laplacian?
2. Relevant equations
[tex]
laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}
[/tex]
3. The attempt at a solution
Is the follow true?
[tex]
\nabla^2\nabla^2=\nabla^4
[/tex]
Also is this true?
[tex]
\nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4}
[/tex]
Or does it include additional terms?
[tex]
\nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4}

+\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial z^2}

+\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial z^2}

+\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial y^2}
[/tex]

Thanks.