Example supplied by Dukemort:
3.1414 squared x9 = 3.365879574290777508693225059404072095292e254
Square rooted x 9 = 3.1414000000000000000000016076. Including calculator errors. further square rooting down to 1.008 etc results in bigger calculator
I think that by this you mean 3.1414^(2^9) which is 3.1414^512. This has 255 digits before the decimal point! And then you want to reverse the process?
How many significant figures are important to you?
In general to maintain the highest accuracy you would need to use a symbolic calculator that works to effectively unlimited precision, like the software Maple or Mathematica.
Without this, limited precision calculators are always going to introduce some rounding errors and these errors will accumulate the more you do repeated calculations. It is generally better to do the calculation in one go rather than do it piecemeal where the errors stand a bigger chance of accumulating.
For example:
The piecemeal result (b) is slightly less accurate. The errors at each stage have fed into the next stage.
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+33667 
