The equation Upper E equals StartFraction 1 Over 2 pi epsilon Subscript 0 EndFraction StartFraction q ⁢ d Over z Superscript 3 EndFraction is approximation of the magnitude of the electric field of an electric dipole, at points along the dipole axis. Consider a point P on that axis at distance z = 23.00d from the dipole center (d is the separation distance between the particles of the dipole). Let Eappr be the magnitude of the field at point P as approximated by the equations below. Let Eact be the actual magnitude. What is the ratio Eappr /Eact?

 E_aprox = 1.003 E_real

Explanation:

In this exercise we are given the expression for the electric field of a dipole in the axis direction of the dipole

         E_real = k 2q d / √(z² + d²)³

I think your equation has some errors.

In this case they indicate that d is the separation of the charges of the dipole

in the case of z »d this equations approximates

        E_aprox = k 2q  d/ z³

calculate the value for the two cases

         E_real = k2q  d / √[ ((23d)² + d²)³]  

         E_real = k2q  d / d³ 12201

         E_real = k2q   1/12201 d²

         E_aprox = k2q   d / (23.00d)³

         E_aprox = k2q   1/12167 d²

    the error between these quantities is

        E_aprox / E_real = 12201 d² / 12167 d²

        E_aprox / E_real = 1.003

        E_aprox = 1.003 E_real

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