Electric field components from a dipole at distant points

[forensics409]

Homework Statement

The problem is: Show that the components of $$\vec{E}$$ due to a dipole are given at distant points, by E_x=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{3pxz}{(x^2+z^2)^{5/2}}$$ and E_z=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{p(2z^2-x^2)}{(x^2+z^2)^(5/2)}}$$


http://physweb.bgu.ac.il/COURSES/PHYSICS2_B/2009A/homework/Homework-2_files/image006.jpg

Homework Equations

E=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{Q}{r^2}$$
p=qd

The Attempt at a Solution

I have tried to break the fields of each one into vector components and add the components, however, it got really messy really quickly and after simplifying it a bit i got a ridiculous equation for just the x component, i had no clue where to go and gave up on even try to get the z component.

E_{x=$$\frac{q}{4\pi\epsilon{o}}$$ [tex]\frac{((x^2+(z+$$\frac{d}{2}$$)^2)^3/2-((x^2+(z-$$\frac{d}{2}$$)^2)^3/2{((x^2+z^2)^2 +($$\frac{x^2d^2}{2}$$-$$\frac{z^2d^2}{2}$$+$$\frac{d^4}{16})^(3/2)$$}[/tex]}

 

[member 553083]

[/tex]I feel the answer should be far simpler than this, but i cannot see what is wrong with my attempt. Any help would be much appreciated.