It would be a very tedious task, but here is how
you could do it.  In fact this is exactly what your 
calculator does when you put in e^(0.135/3) and 
press ENTER or =.

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2.71828 is approximately e

The formula for ex is

1 + x + x2/2! + x3/3! + x4/4! + ··· 

The formula never stops, it keeps adding more and more terms
according to the pattern. The numbers with the exclamation points
are factorials.  4! means 4×3×2×1 or 24 and n! means to multiply n by
every natural number less than it all the way down to 1. 

So you would substitute  into this formula 

1 +  + 2/2 + 3/6 + 3/24 + ···

and take as many terms as you need to get the accuracy you want.

That would be a lot of work, but it could be done, and IS done, by
your calculator.  But it's impossible to ever get the exact answer 
because it is irrational and the decimals never end or repeat in a
pattern.  Maybe this will make you appreciate that a calculator
can do lightning fast what it would take a person hours to do
manually.  I grew up before calculators were invented and we had
to do all calculations manually.  We did have books of tables that 
we used. We used slide rules, too, but they only gave three place
accuracy.  Be glad you live in the electronic age!  :)  

Edwin