We can define a new function by subtracting the two equations gives to get
We want to find the zero's of this function.
Newton's method is given by
This will zero in on each root. Each time you do the process again you will get closer to exact answer.
If you use your seed of 1.6 you will get the accuracy you want after 2 interations.
Let then plug this into the above to get and repeat...
Here is a spread sheet that will allow you to play around with different seed values.