Much larger approximation physics

Question

While reviewing for my EM exam I found that I don't quite understand how to do much larger approximations.

I got the final answer for a Electric Field question as: $$\frac{kq}{(r-a)^2} - \frac{kq}{(r+a)^2}$$

The question asks to make the aproximation: $|r| >> a$.

I don't understand where to go from here. And after checking the answer it isn't my intial idea which was: $$\frac{kq}{(r)^2} - \frac{kq}{(r)^2}$$

Answer 1

Use approximation $(1+\varepsilon)^n\approx1+n\varepsilon$ for very small $\varepsilon$, and write \begin{align} \frac{kq}{(r-a)^2} - \frac{kq}{(r+a)^2} &= \frac{kq}{r^2}\left((1-\frac{a}{r})^{-2}-(1+\frac{a}{r})^{-2}\right)\\ &\approx\frac{kq}{r^2}\left(1+2\frac{a}{r}-1+2\frac{a}{r}\right)\\ &=\dfrac{4akq}{r^3} \end{align} for $a<<r$.

Answer 2

\begin{align}\frac{kq}{(r-a)^2} - \frac{kq}{(r+a)^2} &= kq\frac{4ar}{(r^2-a^2)^2}\\ &\approx\frac{4akqr}{r^4}\\&=\frac{4akq}{r^3}\end{align}

You might want to include the expected answer as it is hard to know how precise you want an approximation to be.