Whats the significance of the $|a|=6$ and $|b|=7$ The problem is as follows:

If $a,b$ and $c$ are group elements and $|a|=6$ and $|b|=7$, express $(a^4c^{-2}b^4)^{-1}$ without negative exponents.

Do I have to rearrange the $(a^4c^{-2}b^4)^{-1}$ someway assuming the group is abelian? 

No need to assume that the group is abelian. Just note that $(xy)^{-1}=y^{-1}x^{-1}$. (Note the reversal.)
In your case, this implies $(a^4c^{-2}b^4)^{-1}=(b^4)^{-1}(c^{-2})^{-1}(a^4)^{-1}=b^{-4}c^2a^{-4}$. Justify the last step and use the hypotheses on the orders to convert negative exponents to positive ones.
